Originally published in ", Science in Context 12 (1998): 137-183.=20

=FDFebruary 14, 1999

The Origins of Eternal Truth in
Modern Mathematics:
Hilbert to Bourbaki and Beyond
Leo Corry1

The Argument:
The belief in the existence of eternal mathematical truth has been part of =
this science throughout history. Bourbaki, however, introduced an interesti=
ng, and rather innovative twist to it, beginning in the mid-1930s. This gro=
up of mathematicians advanced the view that mathematics is a science dealin=
g with structures, and that it attains its results through a systematic app=
lication of the modern axiomatic method. Like many other mathematicians, pa=
st and contemporary, Bourbaki understood the historical development of math=
ematics as a series of necessary stages inexorably leading to its current s=
tate-meaning by this, the specific perspective that Bourbaki had adopted an=
d were promoting. But unlike anyone else, Bourbaki actively put forward the=
 view that their conception of mathematics was not only illuminating and us=
eful to deal with the current concerns of mathematics, but in fact, that th=
is was the ultimate stage in the evolution of mathematics, bound to remain =
unchanged by any future development of this science. In this way, they were=
 extending in an unprecedented way the domain of validity of the belief in =
the eternal character of mathematical truths, from the body to the images o=
f mathematical knowledge as well.=20
Bourbaki were fond of presenting their insistence in the centrality of the =
modern axiomatic method as a way to ensure the eternal character of mathema=
tical truth as an offshot of Hilbert's mathematical heritage. A detailed ex=
amination of Hilbert's actual conception of the axiomatic method, however, =
brings to the fore interesting differences between it and Bourbaki's concep=
tion, thus underscoring the historically conditioned character of certain, =
fundamental mathematical beliefs.


1. Introduction
Throughout history, no science has been more closely associated to the idea=
 of eternal truth than mathematics. This goes without saying. Still, the wa=
y this idea has been conceived has not in itself been eternal and invariabl=
e; rather, it has been the subject of a historical process of development a=
nd change. Yesterday's conception of the eternal character of mathematical =
truth is not identical with today's. Surprisingly, perhaps, this idea has n=
ever been subjected to more serious scrutiny and attack than nowadays.=20
=09In the present article I briefly discuss a short chapter in the long sto=
ry of the development of the idea of eternal truth in mathematics. I will f=
ocus on two central figures that are among the most influential contributor=
s to shaping both the contents of twentieth-century mathematics and our con=
ception of it: Hilbert and Bourbaki. My main point of interest will be thei=
r respective conceptions of the role of the modern axiomatic method in math=
ematics and of its significance concerning the eternal status of mathematic=
al truth. Considering the differences between the two will illustrate the s=
ubtle changes that the status of truth in mathematics may undergo as part o=
f its historical process of development. It will also clarify the backgroun=
d against which current debates on these questions are being held.
=09For the purposes of the present discussion, it is useful to introduce th=
e distinction between the 'body' and the 'images' of scientific knowledge. =
The body of knowledge includes statements that are answers to questions rel=
ated to the subject matter of any given discipline. The images of knowledge=
, on the other hand, include claims which express knowledge about the disci=
pline qua discipline. The body of knowledge includes theories, 'facts', met=
hods, open problems. The images of knowledge serve as guiding principles, o=
r selectors. They pose and resolve questions which arise from the body of k=
nowledge, but which are in general not part of, and cannot be settled withi=
n, the body of knowledge itself. The images of knowledge determine attitude=
s concerning issues such as the following: Which of the open problems of th=
e discipline most urgently demands attention? What is to be considered a re=
levant experiment, or a relevant argument? What procedures, individuals or =
institutions have authority to adjudicate disagreements within the discipli=
ne? What is to be taken as the legitimate methodology of the discipline? Wh=
at is the most efficient and illuminating technique that should be used to =
solve a certain kind of problem in the discipline? What is the appropriate =
university curriculum for educating the next generation of scientists in a =
given discipline? Thus the images of knowledge cover both cognitive and nor=
mative views of scientists concerning their own discipline.
=09The borderline between these two domains is somewhat blurred and it is h=
istorically conditioned. Moreover, one should not perceive the difference b=
etween the body and the images of knowledge in terms of two layers, one mor=
e important, the other less so. Rather than differing in their importance, =
these two domains differ in the range of the questions they address: wherea=
s the former answers questions dealing with the subject matter of the disci=
pline, the latter answers questions about the discipline itself qua discipl=
ine. They appear as organically interconnected domains in the actual histor=
y of the discipline. Their distinction is undertaken for analytical purpose=
s only, usually in hindsight.2
=09Stated in these terms, the issue of the eternal character of mathematica=
l truth is closely connected to the images of mathematical knowledge, since=
 it deals with our conception of the kind of knowledge that mathematics pro=
duces. In the following sections I will discuss the images of mathematics t=
hat underlie the works of Hilbert and of Bourbaki and how they are connecte=
d to the question that occupies us here. Finally, I will also discuss very =
recent debates on the status of mathematical truth, while attempting to pla=
ce these debates in a proper historical perspective.


2. Hilbert
David Hilbert was among the most influential mathematicians of the beginnin=
g of this century, if not the most influential one. The impact of his ideas=
 may be traced up to the present day in fields as distant from one other as=
 number theory, algebraic invariants, geometry, mathematical logic, linear =
integral equations, and physics. His name is often associated with the appl=
ication of the "modern axiomatic approach" to diverse mathematical discipli=
nes and, in the context of the foundations of mathematics, he is usually me=
ntioned as the founder of the formalist school. The term "Hilbert Program,"=
 in particular, refers to the attempt to provide a finitistic proof of the =
consistency of arithmetic, an attempt that G?del's works in the 1930s prove=
d to be hopeless.=20
=09As any German professor educated in the specific intellectual environmen=
t of the end of the nineteenth century, the debates of his philosopher coll=
eagues were not absolutely foreign to Hilbert, and, in fact, his own concep=
tual world was heavily loaded with Kantian and neo-Kantian images. A quotat=
ion of Kant in the frontispiece of his famous Grundlagen der Geometrie (189=
9) is but one, well-known, instance of this. The lecture notes of his cours=
es in G?ttingen contain many more similar examples. There is also abundant =
evidence of his interest and involvement in the careers of philosophers lik=
e Edmund Husserl and Leonard Nelson, and of his hope for a fruitful interac=
tion between them and the G?ttingen mathematicians.3 Likewise, because of h=
is direct involvement in the foundational debates of the 1920s and the infl=
uence of his works in this domain on the subsequent developments of many me=
tamathematical disciplines, his name has pervasively appeared in the contex=
t of twentieth-century discussions about the philosophy of mathematics.
=09But in spite of all this, one has to exercise great care when referring =
to Hilbert's philosophy of mathematics. Over his long years of activity, Hi=
lbert came to deal with many different aspects of mathematics and of physic=
s, facing the development of ever new theories and empirical discoveries an=
d amidst changing historical contexts. Hilbert was fond of making sweeping =
statements about the nature of mathematical knowledge, about the relationsh=
ip between mathematics and science, and about logic and mathematics. These =
statements are abundantly recorded in both published and unpublished source=
s. Sometimes the views expressed in them changed from time to time, if curr=
ent scientific developments demanded so, or if for any other reason Hilbert=
 had changed his mind. And yet the authoritative tone and the total convict=
ion with which Hilbert proclaimed his opinions remained forever the same. C=
oncerning the foundations of physics, for instance, he changed his position=
 around 1913 from a total and absolute support of the idea that all physica=
l phenomena can be reduced to mechanical interactions between rigid particl=
es, to an equally total and absolute defense of an electromagnetic reductio=
nism. He produced important works in fields like the kinetic theory of gase=
s and the general theory of relativity, while holding each of these views r=
espectively. His stress on the importance of each of these positions as a s=
tarting point for physical research is consistently recorded in his lecture=
 notes. Still, neither in his publications nor in his lecture notes one fin=
ds a clue to the fact that his present view was different to the one held b=
efore, nor a word of explanation about the reasons that brought about this =
change of perspective.4
=09As Hilbert was a "working mathematician," whose main professional intere=
sts lie in solving problems, proving theorems, and building mathematical an=
d physical theories, one should not a-priori expect to find any kind of sys=
tematic philosophical discussions in his writings. When these writings do d=
iscuss philosophical issues at all, they often contain claims which are not=
 always supported by solid arguments and which sometimes contradict earlier=
 or later claims. In his interchanges with Gottlob Frege and Luitzen J.E. B=
rouwer, Hilbert even showed a marked impatience with philosophical discussi=
ons. Certainly, it would be misleading to speak about "the philosophy of ma=
thematics of Hilbert," without further qualifications. Instead, it seems to=
 me much more historically illuminating to speak of Hilbert's images of mat=
hematics, and to attempt to elucidate what was more or less steady and perm=
anent in them, on the one hand, and, on the other hand, what changed over t=
ime and under what circumstances. In the present section, I will discuss so=
me of those images, focusing especially on those aspects which are relevant=
 to the use of the axiomatic method and to the question of the provisory or=
 eternal status of mathematical truth.
=09The first publication in which Hilbert thoroughly applied the modern axi=
omatic method was his book Grundlagen der Geometrie (1898). Among Hilbert's=
 sources of inspiration when dealing with the issues covered in this book, =
the most important ones included the German tradition of work on projective=
 geometry (in particular Moritz Pasch's text of 1882), and, perhaps to a so=
mewhat lesser extent, the recent work of Heinrich Hertz on the foundations =
of mechanics.5 The Grundlagen is often read in retrospect as an early manif=
estation of the so-called "formalistic" position, that Hilbert elaborated a=
nd defended regarding the foundations of arithmetic since the 1920s. Under =
this reading, Hilbert conceived geometry as a deductive system in which the=
orems are derived from axioms according to inferences rule prescribed in ad=
vance; the basic concepts of geometry, the axioms and the theorems are  -un=
der this putative conception- purely formal constructs, having no direct, i=
ntuitive meaning whatsoever. This reading of the Grundlagen, however, does =
not reflect faithfully Hilbert's own conception. His approach to geometry, =
at the turn of the century, had a meaningful, empiricist hard-core, in whic=
h the empirical issues of geometry were never lost of sight. In fact, the f=
amous five groups of axioms are so conceived as to express specific, separa=
te ways, in which our intuition of space manifests itself. Hilbert's essent=
ially empiricist conception of geometry is one of those aspects of his imag=
es of mathematics that remained unchanged over the years. The following quo=
tation, taken from the lecture notes of a course taught in K?nigsberg in 18=
91, gives an idea of how he expressed his early conceptions.
Geometry -Hilbert said- is the science that deals with the properties of sp=
ace. It differs essentially from pure mathematical domains such as the theo=
ry of numbers, algebra, or the theory of functions. The results of the latt=
er are obtained through pure thinking ... The situation is completely diffe=
rent in the case of geometry. I can never penetrate the properties of space=
 by pure reflection, much as I can never recognize the basic laws of mechan=
ics, the law of gravitation or any other physical law in this way. Space is=
 not a product of my reflections. Rather, it is given to me through the sen=
ses.6
=09The borderline between those disciplines whose truths can be obtained th=
rough pure thinking and those that arise from the senses sometimes shifted =
in Hilbert's thinking: arithmetic, for instance, is found in his writings i=
n both sides of this borderline at different times. But geometry invariably=
 appears in Hilbert's writing as an empirical science (Hilbert sometimes ev=
en says 'experimental'), similar in essence to mechanics, optics, etc. The =
kind of differences that Hilbert used to stress between the latter and geom=
etry concerned their historical stage of development, rather than their ess=
ence. As he wrote in 1894:
Among the appearances or facts of experience manifest to us in the observat=
ion of  nature, there is a peculiar type, namely, those facts concerning th=
e outer shape of things. Geometry deals with these facts ... Geometry is a =
science whose essentials are developed to such a degree, that all its facts=
 can already be logically deduced from earlier ones. Much different is the =
case with the theory of electricity or with optics, in which still many new=
 facts are being discovered. Nevertheless, with regards to its origins, geo=
metry is a natural science.7
In other words, eventually in the future, when other physical sciences will=
 attain the same degree of historical development than the one geometry has=
 already attained, then there will be no appreciable differences between th=
em, and the kind of axiomatic analysis that one applies now to geometry wil=
l be equally useful for studying other physical sciences.
=09But what is then the meaning of applying a process of axiomatization to =
geometry, one may ask, if this science is in essence an empirical one? The =
aim of the axiomatic analysis that Hilbert presented in the Grundlagen -unl=
ike that of the formalistic conception of axiomatization- was to elucidate =
the logical structure of a given discipline, so that it will become clear w=
hat theorems follow from what assumptions, which assumptions are independen=
t of which, and what assumptions are needed in order to derive the whole bo=
dy of knowledge in that discipline, as we know it at a given stage of its d=
evelopment. In fact, Hilbert's excitement about axiomatization was sparkled=
 by his discovery that a classical technical problem in geometry could be n=
ow overcome, namely, that one does not need infinitesimals in order to reco=
nstruct plane geometry, whereas in space geometry one actually does.8
=09This way of conceiving the role of the axiomatic analysis helps reading =
Hilbert's early works on axiomatization from a perspective which is basical=
ly different from the traditionally accepted one. The question of the consi=
stency of the various kinds of geometries, for instance, which from the poi=
nt of view of Hilbert's later metamathematical research and the development=
s that followed it, might be considered to be the most important one undert=
aken in the Grundlagen, was not even explicitly mentioned in the introducti=
on to that book. Hilbert discussed the consistency of the axioms in barely =
two pages of it, and from the contents of these pages it is not immediately=
 obvious why he addressed this question at all. In 1899 Hilbert did not see=
m to have envisaged the possibility that the body of theorems traditionally=
 associated with Euclidean geometry might contain contradictions, since thi=
s was a natural science whose subject matter is the properties of physical =
space. Hilbert seems rather to have been echoing here an idea originally fo=
rmulated in Hertz's book, according to which the axiomatic analysis of phys=
ical theories will help clearing away possible contradictions brought about=
 over time by the gradual addition of new hypotheses to a specific scientif=
ic theory (Hertz 1894 [1956], 10). Although this was not likely to be the c=
ase for the well-established discipline of geometry, it might still happen =
that the particular way in which the axioms had been formulated in order to=
 account for the theorems of this science led to statements that contradict=
 each other. The recent development of non-Euclidean geometries made this p=
ossibility only more patent. Thus, Hilbert believed that in the framework o=
f his system of axioms for geometry he could also easily show that no such =
contradictory statements would appear.
=09Hilbert established through the Grundlagen the relative consistency of g=
eometry vis-?-vis arithmetic, i.e., he proved that any contradiction existi=
ng in Euclidean geometry must manifest itself in the arithmetical system of=
 real numbers. He did this by defining a hierarchy of fields of algebraic n=
umbers. But in the first edition of the Grundlagen, Hilbert contented himse=
lf with constructing a model that satisfied all the axioms, using only a pr=
oper sub-field, rather than the whole field of real numbers (Hilbert 1899, =
21). It was only in the second edition of the Grundlagen, published in 1903=
, that he added an additional axiom, the so-called "axiom of completeness" =
(Vollst?ndigkeitsaxiom); the latter was meant to ensure that, although infi=
nitely many incomplete models satisfy all the other axioms, there is only o=
ne complete model that satisfies this last axiom as well, namely, the usual=
 Cartesian geometry, obtained when the whole field of real numbers is used =
in the model (Hilbert 1903, 22-24).9
=09The question of the consistency of geometry was thus reduced to that of =
the consistency of arithmetic, but the further necessary step of proving th=
e latter was not even mentioned in the Grundlagen. It is likely that at thi=
s early stage, Hilbert did not yet consider that such a proof could involve=
 a difficulty of principle. Soon, however, he would assign an increasingly =
high priority to it as an important open problem of mathematics.10 Thus, am=
ong the famous list of twenty-three problems proposed by Hilbert in Paris i=
n 1900, the second one concerns the proof of the "compatibility of arithmet=
ical axioms." In formulating this problem, Hilbert articulated his views on=
 the relations between axiomatic systems and mathematical truth, and he thu=
s wrote:
When we are engaged in investigating the foundations of a science, we must =
set up a system of axioms which contains an exact and complete description =
of the relations subsisting between the elementary ideas of the science. Th=
e axioms so set up are at the same time the definitions of those elementary=
 ideas, and no statement within the realm of the science whose foundation w=
e are testing is held to be correct unless it can be derived from those axi=
oms by means of a finite number of logical steps. (Hilbert 1902, 447.)
Views such as this one were at the basis of the well-known debate that aros=
e between Hilbert and Frege immediately after the publication of the Grundl=
agen.11 The latter strongly disputed Hilbert's novel idea, according to whi=
ch logical consistency implied mathematical existence and truth; for Frege,=
 the axioms were necessarily consistent because they were true.12 For Hilbe=
rt, on the other hand, the freedom implied by the possibility of creating n=
ew mathematical worlds based on consistent axiomatic systems was enormously=
 appealing, if not for anything else, for the potential support it seemed t=
o lend to a wholehearted adoption of Georg Cantor's conceptions of the infi=
nite. Moreover, this view endorsed the legitimacy of proofs of existence by=
 contradiction, and thus, a-posteriori, one of Hilbert's early mathematical=
 breakthroughs, namely, his proof of the finite-basis theorem in the theory=
 of algebraic invariants, which had initially encountered with serious diss=
ent by mathematicians of older generations.13
Still, it would be misleading to believe that the mathematical freedom purs=
ued by Hilbert implied a conception of mathematics as a discipline dealing =
with arbitrarily formulated axiomatic systems devoid of any intuitive, dire=
ct meaning. The analysis that Hilbert applied to the axioms of geometry in =
the Grundlagen was based on demanding four properties that need to be met b=
y that system of axioms: completeness,14 consistency, independence, and sim=
plicity. It is true that in principle, there should be no reason why a simi=
lar analysis could not be applied to any other axiomatic system, and in par=
ticular, to an arbitrarily given system of postulates that establishes mutu=
al abstract relations among undefined elements arbitrarily chosen in advanc=
e and having no concrete mathematical meaning. But in fact, Hilbert's own c=
onception of axiomatics did not convey or encourage the formulation of abst=
ract axiomatic systems as such: his work was instead directly motivated by =
the need for better understanding of existing mathematical and scientific t=
heories. In Hilbert's view, the definition of systems of abstract axioms an=
d the kind of axiomatic analysis described above was meant to be carried ou=
t, retrospectively, for 'concrete', well-established and elaborated mathema=
tical entities. In this context, one should notice that in the years immedi=
ately following the publication of the Grundlagen, several mathematicians, =
especially in the USA, undertook an analysis of the systems of abstract pos=
tulates for algebraic concepts such as groups, fields, Boolean algebras, et=
c., based on the application of techniques and conceptions similar to those=
 developed by Hilbert in his study of the foundations of geometry.15 These =
kinds of systems provided an archetype on which Bourbaki eventually modeled=
 the basic definitions of the mathematical structures that constitute in hi=
s view the heart of the various mathematical disciplines. Thus, this is one=
 of the points at which Bourbaki saw his work as a direct continuation of H=
ilbert's intellectual legacy. However, we have no direct evidence that Hilb=
ert showed any interest in the work of the American postulationalist, or in=
 similar undertakings, and in fact there are many reasons to believe that s=
uch works implied a direction of research that Hilbert did not contemplate =
when putting forward his axiomatic program. It seems safe to assert that Hi=
lbert even thought of this direction of research as mathematically ill-conc=
eived.16=20
Hilbert's actual conception of the essence of the axiomatic method is lucid=
ly condensed in the following passage, taken from a 1905 course devoted to =
exposing the principles of the method and its actual application to diverse=
 mathematical and scientific domains:
The edifice of science is not raised like a dwelling, in which the foundati=
ons are first firmly laid and only then one proceeds to construct and to en=
large the rooms. Science prefers to secure as soon as possible comfortable =
spaces to wander around and only subsequently, when signs appear here and t=
here that the loose foundations are not able to sustain the expansion of th=
e rooms, it sets to support and fortify them. This is not a weakness, but r=
ather the right and healthy path of development.17
=09After the publication of the Grundlagen, Hilbert continued to work on th=
e foundation of geometry for the next two-three years, but soon he switched=
 to the next domain of inquiry in which his interests focused over the next=
 period of time: the theory of integral equations. Some of his collaborator=
s in G?ttingen, however, continued to explore the application of the axioma=
tic method to many domains. Thus, for instance, Ernst Zermelo studied in de=
tail the axiomatic foundation of set-theory, while Hermann Minkowski discus=
sed the application of the axiomatic analysis to the latest developments in=
 the electrodynamics of moving bodies. Hilbert followed all these developme=
nts closely, and to a certain extent, actively participated in them.18=20
=09Over time, the issue of consistency became increasingly central to the a=
xiomatic analysis as conceived by Hilbert, especially given the increasing =
centrality of this question to the foundations of arithmetic. Eventually, t=
he requirements of completeness and simplicity of axiomatic systems were pa=
id no more attention, and only independence and consistency of the axioms m=
attered. Simultaneously, the connection between the axiomatic analysis and =
the foundational aspects of mathematics attained more prominence in Hilbert=
's thought. Hilbert continued to relate to the axioms of a given theory as =
historically determined and subject to change, but at the same time he also=
 developed a differentiation between at least two kinds of axioms. This ide=
a was exposed in a now famous lecture held in 1917 in Z?rich, where Hilbert=
 explained the essentials of the axiomatic method as he then conceived it.=
=20
=09Hilbert opened his Z?rich lecture by presenting again the idea that ever=
y elaborated scientific and mathematical theory can be reorganized in such =
a way that its whole body of propositions can be derived from a very limite=
d number of them - the axioms of that theory. Hilbert mentioned many differ=
ent kinds of examples of this situation, among them: the parallelogram law =
as a basic axiom of statics, the law of entropy as a basis for thermodynami=
cs, Kirchhoff's laws of emission and absorption for the theory of radiation=
, Gauss's error law as the basic axiom of the calculus of probabilities, th=
e theorem establishing the existence of roots as basis for the theory of po=
lynomial equations, and -especially interesting for the present discussion-=
 the Riemann conjecture, concerning the purely real character and the frequ=
ency of the roots of the function ((s), as the "foundational law" of the th=
eory of prime numbers (I will return to this example below). All these exam=
ples, by the way, had already been mentioned by Hilbert in many earlier occ=
asions, and he had shown in a more or less detailed fashion how the derivat=
ion of the whole discipline can in fact be realized. The axiomatic derivati=
on of the theory of radiation from the Kirchhoff's law, for instance, const=
ituted an original, and important, contribution of Hilbert, whose publicati=
on attracted much attention (though not always a favorable one).19
=09All these examples, Hilbert explained, illustrate provisory solutions to=
 foundational questions concerning each of the mentioned theories. Very oft=
en in science, however, the need arises to clarify, whether these axioms ca=
n themselves be expressed in terms of more basic propositions belonging to =
a deeper layer. It has been the case, that "proofs" have been advanced of t=
he validity of some of the axioms of the first kind mentioned above: the li=
nearity of the equations of the plane, the laws of arithmetic, the parallel=
ogram law for force-addition, the law of entropy and the theorem of the exi=
stence of roots of an algebraic equation. Hilbert discussed this situation =
in the following terms:
[The] critical test for these "proofs" is manifest in the fact, that they a=
re not themselves proofs, but that at bottom they enable the reduction to d=
eeper-lying propositions which from now on have to be considered as new axi=
oms, instead of the original axioms that we intended to prove. Thus emerge =
what are properly called today axioms of geometry, of arithmetic, of static=
s, of mechanics, of the theory of radiation, or of thermodynamics.... The o=
peration of the axiomatic method, as it has been described here, is thus ta=
ntamount to a deepening of the foundations of the individual scientific dis=
ciplines, very similar to that which eventually becomes necessary while an =
edifice is enlarged and built higher, and we then want to avail for its saf=
ety. (Hilbert 1918, 148. Italics in the original)
=09Hilbert thus stuck to the edifice metaphor as an explanation of the role=
 of the axiomatic method in science, but, at the same time, he laid some st=
ress on the more basic role that certain axioms play from the point of view=
 of foundations. The differentiation suggested here by Hilbert did not expl=
icitly appear in many other places among his writings. It seems to me, howe=
ver, that the ambiguous attitude inherent in this passage implicitly comes =
to the fore in many opportunities, giving rise to diverging interpretations=
 of Hilbert's views according to which of the two aspects, the empirical or=
 the formal, is more strongly stressed. =20

How are these issues related to the question of eternal truths in mathemati=
cs? In the first place, it has already been made clear that Hilbert's inter=
est in the axiomatic method was closely connected with his awareness to the=
 constant changes that scientific theories undergo in the course of their h=
istorical development. This applies to physical as well to mathematical dis=
ciplines. One of the aims of the axiomatic analysis of theories was for Hil=
bert, the possibility of analyzing whether the adoption of new hypothesis i=
nto existing theories would lead to contradiction with the existing body of=
 knowledge, a situation that in his view had been very frequent in the hist=
ory of science. Hilbert thought that the axiomatic analysis of theories cou=
ld help minimizing the appearance of difficulties in the logical structure =
of theories, but certainly not avoid them completely. Still, the question a=
rises how Hilbert thought that open questions at the level of the images of=
 knowledge should be settled, and whether the axiomatic method would play a=
ny role in this, as Bourbaki was later to believe. The answer to this quest=
ion is that Hilbert was somewhat ambiguous towards it.
=09Aware of the power of reflexive mathematical reasoning, Hilbert obviousl=
y thought that some meta-questions about mathematics can be solved within t=
he body of mathematical knowledge, definitely endorsing the answers by mean=
s of standard mathematical proofs. The formalistic program for the foundati=
ons of mathematics, in which he was involved in the 1920s, was based precis=
ely in transforming the very idea of a mathematical proof into an entity su=
sceptible of mathematical study in itself. Occasionally, Hilbert also sugge=
sted that additional meta-questions could also be solved with the help of a=
xiomatic analysis. In his 1917 lecture on axiomatic thinking, Hilbert expla=
ined that a solid foundation for the whole of mathematics would be attained=
 if logic could be properly axiomatized in terms of a consistent system of =
abstract postulates.20 But he also mentioned additional issues, as being cl=
osely related to the latter task:
On closer reflection -Hilbert wrote- we soon recognize that the question of=
 consistency is not an isolated one concerning the integer numbers and the =
theory of sets alone, but rather that it is part of a larger domain of very=
 difficult, epistemological, questions of a specific mathematical hue: in o=
rder to characterize this domain of questions briefly, I mention the proble=
m of the solvability in principle of every mathematical question, the probl=
em of the retrospective controllability of the results of any mathematical =
investigation, then the question of a criterion for the simplicity of a mat=
hematical proof, the question of the relation between contents and formalis=
m in mathematics and in logic, and finally, the problem of the decidability=
 of a mathematical question in a finite number of steps. (Hilbert 1918, 153=
. Italics in the original)
As a matter of historical fact, Hilbert did not himself deal with all these=
 problems in the way he formulated them here. Only the last of these proble=
ms, the Entscheidungsproblem, subsequently became the basis of an actual, f=
ruitful research program. But the specific point I want to make here is tha=
t in instances like this one, Hilbert did discuss the possibility of solvin=
g metamathematical issues inside the body of mathematical knowledge, and mo=
re specifically, with the help of the axiomatic method. Hilbert, however, w=
as aware of the limitations of this approach to solving such issues, and on=
 many occasions he also stressed the contextual, historical or sociological=
, factors that affected the actual answers to questions such as the relativ=
e importance of mathematical theories, or the appropriate way to organize m=
athematical knowledge. In the opening lecture of a course on the foundation=
s of physics, taught in G?ttingen in 1917, Hilbert expressed very clearly t=
his position, while discussing the interrelation of physics and geometry in=
 the aftermath of the development of general relativity. In a passage that =
brings to the fore once again his empiricist view of geometry at a relative=
ly later stage of his career, he said:
In the past, physics adopted the conclusions of geometry without further ad=
o. This was justified insofar as not only the rough, but also the finest ph=
ysical facts confirmed those conclusions. This was also the case when Gauss=
 measured the sum of angles in a triangle and found that it equals two righ=
t ones. That is no longer the case for the new physics. Modern physics must=
 draw geometry into the realm of its investigations. This is logical and na=
tural: every science grows like a tree, of which not only the branches cont=
inually expand, but also the roots penetrate deeper.
Some decades ago one could observe a similar development in mathematics. A =
theorem was considered according to Weierstrass to have been proved if it c=
ould be reduced to relations among integer numbers, whose laws were assumed=
 to be given. Any further dealings with the latter were laid aside and entr=
usted to the philosophers. Kronecker said once: 'The good Lord created the =
integer numbers.' These were at that time a touch-me-not (noli me tangere) =
of mathematics. That was the case until the logical foundations of this sci=
ence began to stagger. The integer numbers turned then into one of the most=
 fruitful research domains of mathematics, and especially of set theory (De=
dekind). The mathematician was thus compelled to become a philosopher, for =
otherwise he ceased to be a mathematician.
The same happens now: the physicist must become a geometer, for otherwise h=
e runs the risk of ceasing to be a physics and viceversa. The separation of=
 the sciences into professions and faculties is an anthropological one, and=
 it is thus foreign to reality as such. For a natural phenomenon does not a=
sk about itself whether it is the business of a physicits or of a mathemati=
cian. On these grounds we should not be allowed to simply accept the axioms=
 of geometry. The latter may be the expression of certain facts of experien=
ce that further experiments would contradict.21

=09The 1920s are the years of Hilbert's more intense involvement with the q=
uestions of foundations. But even in this period there is plenty of evidenc=
e that his basic views on the place of uncertainty in mathematics did not c=
hange in any essential way. In 1919-20, Hilbert gave a series of popular le=
ctures in G?ttingen under the general title of "Nature and Mathematical Kno=
wledge." In these lectures Hilbert sharply criticized accepted views of mat=
hematics and physics. He explicitly discarded the view that mathematics can=
 be reduced to a formal game played with meaningless symbols according to r=
ules established in advance, while stressing the role of intuition and expe=
rience as a source of mathematics. He also discussed the place of conjectur=
al thinking, and the fallibility of mathematical reasoning. Of particular r=
elevance for the present discussion is the connection that Hilbert establis=
hed between the axiomatic method and conjectural thinking in mathematics.=
=20
=09It is clear that any axiomatically developed theory has an hypothetical =
character, in the sense that the conclusions of the theory are valid whenev=
er the validity of the axioms is assumed. For instance, in every mathematic=
al situation in which the conditions of the elementary axioms of the theory=
 of rings are satisfied, the theory provides certain theorems of unique fac=
torization that are valid in that situation. This is the way in which Bourb=
aki, as we will see, presented the essence of the axiomatic method, and cle=
arly this description applies to a large extent to Hilbert's view as well. =
But in these lectures of 1919-20, we also find a broader conception of the =
application of the method as Hilbert conceived it: the possibility of incor=
porating into the body of mathematical knowledge theories that are based on=
 unproved, tough perhaps plausible, theorems of significant content.
=09Hilbert had in mind, in this case, the particular example of the already=
 mentioned Riemann conjecture. From the point of view of the calculus of pr=
obability, this conjecture certainly appears, a-priori, as a rather implaus=
ible one, since it demands that the zeros of a certain function will all li=
e on a very delimited region of space. Still, what we know about mathematic=
s, and in particular, our knowledge of the fruitful results that seem to fo=
llow from this conjecture lead us to assign it a high plausibility of being=
 true. Moreover, Hilbert saw it as legitimate to build a full mathematical =
theory based on the assumption of the validity of this conjecture, but only=
 insofar as a correct application of the axiomatic method will help us keep=
ing track of the limitations of such a theory. Hilbert repeated in these le=
ctures many of the ideas exposed in 1917 in Z?rich, and in particular the i=
dea of two different layers of axioms, bearing a different foundational cha=
racter. Echoing this distinction, Hilbert explained the possible use that c=
an be made of conjectures such as Riemann's. Thus Hilbert wrote:=20
In discussing the method of mathematics, I have already stressed that when =
building a particular theory, it is a fully justified procedure to assume s=
till unproved, but plausible, theorems (as axioms), provided one is clear a=
bout the incomplete character of this way of laying the foundations of the =
theory. (Hilbert 1919-20 [1992], 78)
=09In 1922-23 Hilbert gave another series of popular lectures in G?ttingen.=
 Among other topics he discussed the place of error in the history of mathe=
matics. Hilbert plainly declared that errors had played a significant role =
in the development of this science. The passage where he explains his view =
presents an image of how progress is attained in mathematics, which, again,=
 is totally opposed to the one Bourbaki tried so hard to put forward more t=
han two decades later. I quote Hilbert in some extension:
Every time that a new, fruitful method is invented in order to solve a prob=
lem, in order to expand our knowledge, or in order to conquer new provinces=
 of science, there are, on the one hand, critical researchers who distrust =
the novelty, and on the other hand, the courageous ones, who before all oth=
ers deplete the inexhaustible and productive source, swiftly achieve innova=
tion and soon even gain overweight of it, so that they can silence the obje=
ctions of the critics. This is the period of the swift advancement of scien=
ce. Often the best pioneers are those who dare to advance deeper and are th=
e first to arrive to unsafe territory. Signs of the latter are unclearness =
and uncertainty in the results obtained, to the point that even visible con=
tradictions and countersenses -the so-called paradoxes- arise. At this mome=
nt reappear on the stage the critical tendencies, that until now have stood=
 aside. They take possession of the paradoxes, uncover real mistakes and th=
us attempt to incriminate the whole method and to reject it. The danger exi=
sts that all the progress achieved will be lost. The main task in such a si=
tuation is to hold this criticism back (einzud?mmen) and to look after a re=
formulation of the foundations of the method, so that it remains safe from =
all its false applications and, at the same time, that the ordinary results=
 of the established portions of mathematical knowledge can be incorporated =
into it. (Hilbert 1922-23, 38-39)
In the past, Hilbert had himself played the role of a courageous pioneer in=
 the framework of his early work on algebraic invariants. In 1888 he proved=
 the existence of a finite basis for every system of invariants using new m=
ethods that were, at first, harshly criticized by more conservative mathema=
ticians. In 1922, he was clearly referring to his current concern with the =
foundations of mathematics. From very early on Hilbert had defended the new=
 conception of the infinite implied by the work of Cantor on sets, and his =
formalist program was conceived as way of countering the criticism of those=
 who thought that the acceptance of the actual infinite in mathematics was =
damaging, as the appearance of paradoxes suggested. But in any case, Hilber=
t's open-minded attitude towards alternative and innovative views in mathem=
atics, highly contrasts with that of Bourbaki, as will be described below. =
And again, Hilbert did not believe that the axiomatic conception of mathema=
tical theories would totally safeguard against error. =20
 =09Hilbert's conception of the axiomatic method and of its role in science=
, then, contained various, somewhat diverging, elements. Small wonder, then=
, that different mathematicians at different times derived different ideas =
from it. Clearly, Bourbaki's conceptions are elaboration of some of these e=
lements, but leave aside others. But even among Hilbert's students and coll=
eagues, and still during his lifetime, one can observe how these various el=
ements are differently stressed.
=09On the occasion of Hilbert's sixtieth birthday, the journal Die Naturwis=
senschaften dedicated one of its issues to celebrate the achievements of th=
e master. Several of his students were commissioned with articles summarizi=
ng Hilbert's contributions in different fields. Max Born, who as a young st=
udent in G?ttingen attended many of Hilbert's courses, and later on as a co=
lleague continued to participate in his seminars, wrote about Hilbert's phy=
sics. Born was perhaps the physicist that expressed a more sustained enthus=
iasm for Hilbert's physics. He seems also to have truly appreciated the exa=
ct nature of Hilbert's program for axiomatizing physical theories and the p=
otential contribution that the realization of that program could yield. His=
 description of the essence of this program stressed its empiricist underpi=
nnings, and at the same time attempted to explain why, in general, physicis=
ts tended not to appreciate it. Curiously, he directly addressed the issue =
of the relationship between the modern axiomatic method and eternal mathema=
tical truth. Born put it in the following words:
The physicist set outs to explore how things are in nature; experiment and =
theory are thus for him only a means to attain an aim. Conscious of the inf=
inite complexities of the phenomena with which he is confronted in every ex=
periment, he resists the idea of considering a theory as something definiti=
ve. He therefore abhors the word "Axiom", which in its usual usage evokes t=
he idea of definitive truth. The physicist is thus acting in accordance wit=
h his healthy instinct, that dogmatism is the worst enemy of natural scienc=
e. The mathematician, on the contrary, has no business with factual phenome=
na, but rather with logic interrelations. In Hilbert's language the axiomat=
ic treatment of a discipline implies in no sense a definitive formulation o=
f specific axioms as eternal truths, but rather the following methodologica=
l demand: specify the assumptions at the beginning of your deliberation, st=
op for a moment and investigate whether or not these assumptions are partly=
 superfluous or contradict each other. (Born 1922, 591)
   =09In Born's view, then, Hilbert's axiomatic approach is not applied in =
order to attain eternal truth, as Bourbaki's views later implied, but rathe=
r in order to enable a clearer understanding of the nature of our provisory=
 conceptions, and in order to provide the means to correct errors that migh=
t arise in them. But in the same issue of that journal dedicated to Hilbert=
, a somewhat different (although not contradictory) assessment of his work =
appears, in an article by Paul Bernays on "Hilbert's Significance for the P=
hilosophy of Mathematics." Bernays was at that time Hilbert's closest assis=
tant, and together they dedicated most of their current efforts to foundati=
onal questions, and, in particular, to lay down the basis for the realizati=
on of the so-called "Hilbert Program." Clearly, when presenting Hilbert's i=
deas, Bernays stressed mainly those connected with his foundational concern=
s. Bernays explained the essence of Hilbert's axiomatic conception, and cla=
imed that the main task of his analysis was the proof of consistency of the=
 theories involved. This was certainly true for Hilbert's current concerns,=
 but, as I claimed above, it had been much less the case for his early inve=
stigations of the foundations of geometry and of physics. From the vantage =
point of view of later developments, Bernays saw this as a constant point o=
f major interest for Hilbert. After explaining the motivation for Hilbert's=
 current interest in investigating the nature of mathematical proofs, Berna=
ys explained the philosophical meaning of the master's entire endeavor, in =
the following terms:
While clarifying the workings of mathematical logic, Hilbert transformed th=
e meaning of this method [of the logical calculus] in a way very similar, t=
o that earlier applied for the axiomatic method. Very much like he had once=
 striped off the visualizable (Anschauulich) contents out of the basic rela=
tions and of the axioms of geometry, so he detached now the mental contents=
 of deductions from the proofs of arithmetic and analysis, which he had mad=
e the subject-matter of his investigations. He did so by taking as his imme=
diate object of consideration the systems of formulae through which those p=
roofs are represented in the logical calculus, cut off from any logical-con=
tentual interpretation. In this way he could substitute the methods of proo=
fs used in analysis with purely formal transactions, which are performed on=
 determinate signs according to fixed rules.
By means of this approach, in which the separation of what is specifically =
mathematical from anything that has to do with contents reaches its peak, t=
he Hilbertian conception of the essence of mathematics and of the axiomatic=
 method attains for the first time its true realization. For we recognize f=
rom now on, that the sphere of the mathematical-abstract into which the mat=
hematical way of thinking translates all what is theoretically conceivable,=
 is not the sphere of what has logical content, but rather it is the domain=
 of the pure formalism. Mathematics thus appears as the general theory of f=
ormal systems (Formalismen), and, since we are able to conceive it that way=
, also the universal meaning of this science becomes clear at once. (Bernay=
s 1922, 98)
As we will see below, Bernays' characterization of Hilbert's axiomatic meth=
od is very close to the one accorded to him by Bourbaki, and to the one put=
 forward in Bourbaki's mathematics. This is also the one that has come to b=
e more closely associated with Hilbert's name. But as we have seen, this is=
 only one aspect of Hilbert's much more complex conception, and an excessiv=
e stress on it runs the risk of leading to misinterpretation: this is parti=
cularly the case when it comes to the link between the axiomatic method and=
 eternal truth in mathematics.

3. Bourbaki
I proceed to discuss now the work of Nicolas Bourbaki and the conception of=
 the status of mathematical truth put forward in this work. Nicolas Bourbak=
i is the pseudonym adopted in the mid-1930s by a group of young French math=
ematicians, who undertook the task of collectively writing an up-to-date tr=
eatise of mathematical analysis, suitable both as a textbook for students a=
nd as a source of reference for researchers. The founding members of the gr=
oup were initially motivated by an increasing dissatisfaction with the text=
s of mathematical analysis currently used in their country, and by a feelin=
g that French mathematical research was lagging far behind that of other co=
untries, especially Germany. The project materialized in a way that perhaps=
 the members of the group had not truly anticipated, and the published trea=
tise -which covers many fields of modern mathematics, rather than analysis =
alone- became one of the most influential texts of twentieth-century mathem=
atics: the El?ments de Math?matique. Among the many ways that Bourbaki's in=
fluence was felt, most relevant for the present discussion is the entrenchm=
ent and broadening of the idea that mathematics deals with eternal truths.=
=20
=09The name of Bourbaki has been associated more than any other one to a ve=
ry influential and pervasive image of twentieth-century mathematics, namely=
, the idea that mathematics is a science of "structures."22 Volume One of t=
he Elements deals with the theory of sets, and its fourth chapter introduce=
s and discusses the concept of structure.23 Allegedly, this formally define=
d mathematical concept is meant to provide the solid foundation on which th=
e whole picture of mathematics put forward by the treatise will be built. T=
his is a unified and, on the face of it, very coherent picture in which mat=
hematics is seen as a hierarchy of structures of increasing complexity. Acc=
ording to Bourbaki's image, the aim of mathematical research is to elucidat=
e the essence of each of these structures.=20
=09However, the actual place of the idea of structure in Bourbaki's mathema=
tics is much more complex than what this simple -and very often accepted wi=
th uncritical scrutiny- account seems to imply. The same term "structure" i=
s used in Bourbaki's texts, rather indiscriminately, with two different mea=
nings. One the one hand, there is the above mentioned, formal concept of st=
ructure. On the other hand, there is a more general, undefined and non-form=
al idea of what a "mathematical structure" is. Bourbaki's theory of structu=
res is hardly used in developing the theories that Bourbaki included in the=
 treatise, and where it does appear, it can absolutely be dispensed with. O=
utside Bourbaki's treatise, moreover, structures were ever mentioned in les=
s than a handful of instances by mathematicians. On the contrary, the struc=
tural conception of mathematics, understood as a non-formally conceived ima=
ge of mathematical knowledge, proved extremely fruitful for Bourbaki's own =
work, and at the same time exerted a profound influence on generations of m=
athematicians all around the world.24=20
=09The roots of both senses of the term "structure," and the details regard=
ing the way in which they were used by Bourbaki and by their followers will=
 not concern us in the present article. Neither will we discuss the extent =
of the group's influence and how it contributed to shaping the course of de=
velopment of mathematics over several decades of the present century. The f=
ocus of our attention will concentrate on the interrelations between struct=
ures, axiomatic thinking and eternal truth in Bourbaki's conception of math=
ematics, and on the alleged influence of Hilbert's ideas on these issues. T=
he main claim I want to stress here is that Bourbaki's introduction of  the=
 concept of structure can be explained in terms of the group's images of ma=
thematics: Bourbaki's account of mathematics in terms of "structures" was a=
n attempt to extend the idea that mathematics produces eternal truths, from=
 the domain of the body of mathematics (in which it is commonly accepted), =
to include also the images of mathematics. I proceed to discuss, then,  Bou=
rbaki's images of mathematics and the way how the idea of "structure", as w=
ell as the theory of structures, are connected with them.
=20
Although terms like "Bourbaki's philosophy of mathematics" or "Bourbaki's s=
tructuralist program for mathematics," are very frequently used, it is rath=
er uncommon to find a detailed account of what these terms mean. Morris Kli=
ne, for instance, described this program as follows:
Nicolas Bourbaki undertook in 1936 to demonstrate in great detail what most=
 mathematicians believed must be true, namely, that if one accepts the Zerm=
elo-Fraenkel axioms of set-theory, in particular Bernay's and G?del's modif=
ication, and some principles of logic, one can build up all of mathematics =
on it. (Kline 1980, p. 256)
Isolated passages like this one abound, but they are both inaccurate and ve=
ry partial as a description of what Bourbaki's undertaking amounted to. Som=
e more articulate attempts to explain it (e.g., Fang 1970, Stegm?ller 1979)=
, do not throw much light on the issue either.
=09As a matter of fact, a consistent and systematic program, clearly formul=
ated and generally accepted by all members of the group, was never at the b=
asis of Bourbaki's work. As I said above, the group's work began as an atte=
mpt to write a new treatise on analysis, and it was only in the process of =
writing that its scope was broadened to include many other fields of mathem=
atics. Also the austere axiomatic style that characterizes this work was ad=
opted only progressively. Moreover, as I have shown elsewhere, the very ide=
a of attributing such a centrality to the notion of "structure" for their p=
resentation of mathematics arose relatively late, it was never fully adopte=
d by all members as a leading principle, and, in fact, it proved quite prob=
lematic. Very much like Hilbert before them, the members of the group saw t=
hemselves, above all, as "working mathematicians", focused on activities su=
ch as problem solving, research and exposition of theorems and theories. Th=
eir interest in philosophical or foundational issues was only oblique, and =
certainly not systematic. Bourbaki never formulated an explicit philosophy =
of mathematics and, in retrospect, individual members of the group even den=
ied any interest whatsoever in philosophy or even in foundational research =
of any kind.
=09And yet, it is nevertheless possible to reconstruct what, in retrospect =
can be defined as Bourbaki's images of mathematics, as a way to understand =
the context of their mathematical activity. Obviously, this system of image=
s of knowledge, is one that was subject to constant criticism (both externa=
l and internal), that evolved through the years, and, that, occasionally, i=
nvolved ideas that were in opposition to the actual work whose setting they=
 were meant to provide. This can be said, in fact, of the images of knowled=
ge of every individual scientist, but in the case of Bourbaki, a group that=
 gathered together various leading mathematicians with strong opinions abou=
t every possible issue, all these factors need to be more strongly stressed=
. In fact, one has to take in account that, very often throughout their man=
y years of activity, members of the group professed conflicting beliefs at =
the level of the images of knowledge.=20
=09Taking in account all these necessary qualifications, Bourbaki's images =
of mathematics can be reconstructed by directly examining the mathematical =
work and the historical accounts of the development of mathematics publishe=
d by the group, by examining pronouncements of different members of the gro=
up, and from several other sources as well. Jean Dieudonn? has no doubt bee=
n the most outspoken member of the group. More than anyone else, he was res=
ponsible for producing and spreading the popular conception of what Bourbak=
i's mathematics are. The views of the majority of the group's members -in p=
articular, those views concerning the structural conception of mathematics =
and the role of the concept of structure in the work of Bourbaki- have been=
 usually much less documented or not documented at all.=20
=09Bourbaki began its work amidst a multitude of newly obtained results, so=
me of them belonging to branches of mathematics that were only incipient. T=
he early years of Bourbaki's activity witnessed a boom of unprecedented sco=
pe in mathematical research. In 1948 Dieudonn? published, signing with the =
name of Bourbaki, a now famous article that was later translated into sever=
al languages and which has ever since come to be considered the group's pro=
grammatic manifesto: "The Architecture of Mathematics." According to the pi=
cture of mathematics described in that article, the boom in mathematical re=
search at the time of its writing raised the pressing question, whether it =
could still be legitimate to talk about a single discipline called "mathema=
tics", or:
... whether the domain of mathematics is not becoming a tower of Babel, in =
which autonomous disciplines are being more and more widely separated from =
one another, not only in their aims, but also in their methods and even in =
their language. (Bourbaki 1950, 221)
Dieudonn? stressed the role of the axiomatic method as an underlying common=
 basis for a unified view of mathematics, in face of its apparent disunity.=
 Thus Dieudonn? wrote:
Today, we believe however that the internal evolution of mathematical scien=
ce has, in spite of appearance, brought about a closer unity among its diff=
erent parts, so as to create something like a central nucleus that is more =
coherent than it has ever been. The essential part of this evolution has be=
en the systematic study of the relations existing between different mathema=
tical theories, and which has led to what is generally known as the "axioma=
tic method." ... Where the superficial observer sees only two, or several, =
quite distinct theories, lending one another "unexpected support" through t=
he intervention of mathematical genius, the axiomatic method teaches us to =
look for the deep-lying reasons for such a discovery. (Bourbaki 1950, 222-2=
23)
According to Dieudonn?, then, the modern axiomatic method lies at the heart=
 of mathematics, and it is precisely the use of this method what allows to =
preserve its unity. I will call this idea "the axiomatic image of mathemati=
cs", i.e., the idea that mathematics is the science dealing with axiomatic =
systems.
=09It is interesting to notice that in his Paris address of 1900, Hilbert h=
ad already manifested his concern with the possible danger of internal dism=
emberment of mathematics, given the current diversity among its sub-discipl=
ines. Hilbert expressed himself in terms similar to those later used by Die=
udonn? in 1948, which may be more than a simple coincidence. Thus Hilbert s=
aid:
The question is urged upon us whether mathematics is doomed to the fate of =
those other sciences that have split up into separate branches, whose repre=
sentatives scarcely understand one another and whose connections become eve=
r more loose. I do not believe it nor wish it. Mathematical science is in m=
y opinion an indivisible whole, an organism whose vitality is conditioned u=
pon the connection of its parts. For with all the variety of mathematical k=
nowledge, we are still clearly conscious of the similarity of the logical d=
evices, the relationship of the ideas in mathematical theory and the numero=
us analogies in its different departments. We also notice that, the farther=
 a mathematical theory is developed, the more harmoniously and uniformly do=
es its construction proceed, and unsuspected relations are disclosed betwee=
n hitherto separate branches of the science. (Hilbert 1902, 478-479)
But between Hilbert's 1900 address and Dieudonn?'s manifesto had elapsed al=
most half a century, and the problem of the (dis-)unity of mathematics was =
more pressing than ever before. At the same time, the modern axiomatic meth=
od had become a mainstream language of many mathematical branches, and in a=
 certain sense (which we will discuss immediately), it had departed from Hi=
lbert's initial conception. Still, what exactly the application of the mode=
rn axiomatic method amounts to, and what is the meaning of this for mathema=
tics, was not a straightforward issue even among Bourbaki members. Thus, He=
nri Cartan, one of the founding members of Bourbaki, defined it as follows:
A mathematician setting out to construct a proof has in mind well defined m=
athematical objects which he is investigating at the moment. When he thinks=
 he has found the proof, and begins to test carefully all his conclusions, =
he realizes that only a very few of the special properties of the objects u=
nder consideration played a role in the proof at all. He thus discovers tha=
t he can use the same proof for other objects which have only those propert=
ies he had employed previously. Here we can see the simple idea underlying =
the axiomatic method: instead of declaring which objects are to be investig=
ated, one has to list those properties of the objects to be used in the inv=
estigation. These properties are then brought to the fore expressed by axio=
ms; whereupon it ceases to be important to explain what these objects are, =
that are to be studied. Instead, the proof can be constructed in such a way=
 as to hold true for every object that satisfies the axioms. It is quite re=
markable how the systematic application of such a simple idea has shaken ma=
thematics so completely. (Cartan 1958 [1980], 176-177. Italics in the origi=
nal)
This description of the essence of the axiomatic method is not only simple =
and clear; it is also very similar to the one intended by Hilbert when he f=
irst applied it to the foundations of geometry. However, influenced by late=
r works of Hilbert, particularly by his works on the foundations of logic a=
nd arithmetic, different aspects of the axiomatic method came to be stresse=
d more strongly, as I already suggested, and a somewhat different picture a=
rose. This is the picture that relates Hilbert to the formalist approach of=
 mathematics, of which Bourbaki became a leading promoter. Dieduonn?, for i=
nstance, described Hilbert's conception by comparing it to a game of chess.=
 In the latter one does not speak about truths, but rather about following =
correctly a set of stipulated rules. If we transpose that idea into mathema=
tics -wrote Dieudonn?- we arrive to Hilbert's conception: mathematics becom=
es a game, in which the pieces are graphical signs, that are distinguished =
from each other by their form alone, and not by their contents (Dieudonn? 1=
962, 551).
=09Dieudonn?'s rendering of Hilbert's position, though not absolutely faith=
ful as an historical description, is the one that more faithfully describes=
 the kind of mathematics that developed under the influence of Bourbaki's o=
wn textbooks. These books present the various domains discussed on them as =
defined by a list of apparently meaningless axioms, and all the results are=
 derived with reference only to these axioms, while explicitly excluding an=
y kind of motivation or intuition. The trademark of these texts is that the=
y exclude any external references (though there are many cross-references) =
as well as any reliance on figures, even in domains which are so strongly g=
eometrically motivated as topology. The axiomatic image of mathematics more=
 closely associated to the name of Bourbaki -and the one I will refer to he=
re- is the image according to which mathematics is a series of formal theor=
ies, at the basis of which stand axioms without any specific, intuitive mea=
ning, and the results of which are supported by formally constructed proofs=
 without any appeal to external intuitions. When Bourbaki has been declared=
 the "legitimate heir of Hilbert", this image of mathematics has, by implic=
ation, been also associated very often with Hilbert himself. =20

The axiomatic image of mathematics is, then, the first basis on which Bourb=
aki's conception of mathematics is built. Closely associated with it, but n=
onetheless different, is "the structural image of mathematics," according t=
o which the object of mathematical research is the elucidation of the vario=
us structures that appear in it. The first full-fledged realization of this=
 image was put forward in a classical textbook published in 1930 by the Dut=
ch mathematician B.L. van der Waerden, under the name of Moderne Algebra (1=
930). Building on ideas he had learnt as a student of Emmy Noether and of E=
mil Artin, van der Waerden presented the various branches of this mathemati=
cal domain under the leading notion that all of them are manifestations of =
a single, unifying idea, namely, that algebra is the discipline dealing wit=
h the study of the various algebraic structures.25 Deeply impressed by van =
der Waerden's achievement in algebra, Bourbaki undertook to present much la=
rger portions of mathematics in a similar way. Dieudonn? described the unif=
ying role of the structures as follows:
Each structure carries with it its own language, freighted with special int=
uitive references derived from the theories from which the axiomatic analys=
is ... has derived the structure. And, for the research worker who suddenly=
 discovers this structure in the phenomena which he is studying, it is like=
 a sudden modulation which orients at once the stroke in an unexpected dire=
ction in the intuitive course of his thought and which illumines with a new=
 light the mathematical landscape in which he is moving about.... Mathemati=
cs has less than ever been reduced to a purely mechanical game of isolated =
formulas; more than ever does intuition dominate in the genesis of discover=
ies. But henceforth, it possesses the powerful tools furnished by the theor=
y of the great types of structures; in a single view, it sweeps over immens=
e domains, now unified by the axiomatic method, but which were formerly in =
a completely chaotic state. (Bourbaki 1950, 227-228)
Thus Dieudonn? attributed to the structures -and especially to "the theory =
of the great types of structures"- a central role in the unified picture of=
 mathematics. This was an innovative idea, based on which Bourbaki was able=
 to exert a deep influence on future research.=20
=09Van der Waerden defined all the algebraic structures studied in his book=
 in strict axiomatic terms. Bourbaki did the same for the structures invest=
igated in the various volumes of the El?ments. Still, the fact must be stre=
ssed that the axiomatic image and the structural image are different ideas;=
 a significant manifestation of this is that a notion of mathematical struc=
ture, similar to that of van der Waerden or of Bourbaki, is absent from Hil=
bert's work, in spite of the centrality played in the latter by the axiomat=
ic image of mathematics. In fact, comparing the role played by the idea of =
structure in Moderne Algebra, on the one hand, and in Bourbaki and in Hilbe=
rt respectively, on the other hand, sheds additional light on the different=
 roles that Hilbert and Bourbaki assigned to the place of axiomatically def=
ined theories in their whole conception of mathematics.
As already mentioned, van der Waerden's is the first paradigmatic book in w=
hich a mathematical discipline, algebra, was presented as a science of stru=
ctures. Moderne Algebra profoundly influenced the conceptions of the foundi=
ng members of Bourbaki, and their whole endeavor may be considered, to a la=
rge extent, as an attempt to extend this view from the relatively limited s=
cope of algebra alone to mathematics at large. On the other hand, van der W=
aerden himself worked in G?ttingen in 1927 with Emmy Noether, and her cours=
es provided a main source of inspiration for his book. These circumstances =
might tend to support the view that there is a clear thread connecting Hilb=
ert's basic conceptions of the nature of mathematics to those of Bourbaki. =
On closer examination, however, a different picture arises.
=09Van der Waerden's book appeared around the time of Hilbert's retirement.=
 We have no direct evidence of what was his opinion of the book and of the =
image of algebra presented there. We do know, however, that during his life=
time, Hilbert never taught courses on the issues that constitute the hard c=
ore of van der Waerden's presentation of algebra. The abstract theory of ri=
ngs, for instance, and its use of the abstract concept of ideal as a main t=
ools for studying unique factorizations in he most general terms, were abso=
lutely recent developments, that Hilbert never used in its newer formulatio=
n. The abstract fields that van der Waerden used in his work were very diff=
erent from the concrete fields of algebraic numbers that Hilbert had thorou=
ghly analyzed in his own work. The theory of abstract groups was a relative=
ly well elaborated mathematical domain by the turn of the century, but Hilb=
ert never showed a specific interest in it, independently of its applicatio=
n to other, more classical domains of nineteenth-century mathematics. Moreo=
ver, among the sixty eight dissertations that Hilbert supervised in his lif=
etime, none of them deals with these kinds of issues. Likewise, although fi=
ve among the twenty-three problems that Hilbert included in his 1900 list c=
an be considered in some sense as belonging to algebra in the nineteenth-ce=
ntury sense of the word, none of them deals with problems connected with mo=
re modern algebraic concerns, and in particular not with the theory of grou=
ps.
=09Hilbert's mathematical work surely implied important innovations at many=
 levels, in both the body and the images of knowledge, but at the same time=
, it had deep roots in the classical domains of nineteenth century mathemat=
ics, and in the views associated with them. One of these view concerned the=
 foundational status of the various systems of numbers. Under the classical=
, nineteenth-century image of mathematics these systems lie at the heart of=
 all mathematical knowledge, and algebra is built on top of them. Hilbert c=
ontributed to the elaboration of many new tools for the study of algebra an=
d of the systems of numbers, but he never changed the traditional conceptua=
l hierarchy. In van der Waerden's structural image of algebra, the hierarch=
y is totally reversed, and these tools, defined by means of abstract system=
s of postulates transform into the basic mathematical entities. The basic s=
ystems of numbers (integers, rationals, reals) become particular cases of m=
ore general algebraic structures.=20
How did Hilbert reacted to this change in the conceptual order, in which th=
e system of real numbers is dependent on the results of algebra rather than=
 being the basis for it? Based on what we know about Hilbert's images of ma=
thematics, it seems safe to conjecture that his attitude in this respect ma=
y have been ambiguous at best. An indication of what this attitude may have=
 been appears in Hermann Weyl's obituary to Hilbert. Regarding Hilbert's co=
nception of the role of axiomatics in modern algebra Weyl stated:
Hilbert is the champion of axiomatics. The axiomatic attitude seemed to him=
 one of universal significance, not only for mathematics, but for all scien=
ces. His investigations in the field of physics are conceived in the axioma=
tic spirit. In his lectures he liked to illustrate the method by examples t=
aken from biology, economics, and so on. The modern epistemological interpr=
etation of science has been profoundly influenced by him. Sometimes when he=
 praised the axiomatic method he seemed to imply that it was destined to ob=
literate completely the constructive or genetic method. I am certain that, =
at least in later life, this was not his true opinion. For whereas he deals=
 with the primary mathematical objects by means of the axioms of his symbol=
ic system, the formulas are constructed in the most explicit and finite man=
ner. In recent times the axiomatic method has spread from the roots to all =
branches of the mathematical tree. Algebra, for one, is permeated from top =
to bottom by the axiomatic spirit. One may describe the role of axioms here=
 as the subservient one of fixing the range of variables entering into the =
explicit constructions. But it would not be too difficult to retouch the pi=
cture so as to make the axioms appear as the masters. An impartial attitude=
 will do justice to both sides; not a little of the attractiveness of moder=
n mathematical research is due to the happy blending of axiomatic and genet=
ic procedures. (Weyl 1944, 645)
=09The kind of impartial attitude promoted here by Weyl, and which probably=
 was very close to Hilbert's original views, is not really the attitude tha=
t the images of mathematical knowledge put forward in Bourbaki's treatise m=
ake manifest. Whereas in Bourbaki's structural image, the axiomatic systems=
 lie at the basis of mathematics and are the starting point for the develop=
ment of theories, for Hilbert, the axiomatic analysis is a relatively late =
stage in it. Hilbert's axiomatic analysis is part of an open-ended, flexibl=
e and mainly empirically motivated process of knowledge-creation in mathema=
tics, rather than the origin and justification of a rigidly conceived, and =
a-priori determined course of evolution, that is realized by means of logic=
al deduction alone.

Bourbaki, then, attempted to extend the structural image from algebra to th=
e whole of mathematics. But beyond the different scopes of these two enterp=
rises, there is an additional, much more significant difference between Bou=
rbaki's and van der Waerden's structural images of mathematics, and between=
 the notions of structure underlying each of them. Van der Waerden never pr=
ovided an explicit explanation, either formal or non-formal, of what is to =
be understood by an "algebraic structure" or by "structural research in alg=
ebra"; he showed what this is by simply doing it. Bourbaki, unlike van der =
Waerden in this respect, not only attempted on various opportunities to exp=
lain what the structural approach is and why it is so novel and important f=
or mathematics, but, moreover, they formulated what they expected to be an =
elaborate mathematical theory, the theory of structures, meant to sustain a=
nd endorse their explanations -and in fact their whole system of images of =
mathematics- by means of an allegedly unifying, mathematical theory.=20
=09This attempt is connected to a third basic element -together with the ax=
iomatic image and the structural image- of Bourbaki's conception of mathema=
tics: the reflexive image. In contrast with other exact sciences, mathemati=
cal knowledge displays the peculiarity of enabling -within the body of math=
ematics properly said and using similar standards of proof and similar tech=
niques to those used in any other mathematical domain- the study and elucid=
ation of particular aspects of that system of knowledge constituted by math=
ematics. In other words: mathematics affords the possibility of reflexively=
 studying, within the body of knowledge, certain issues belonging to the im=
ages of knowledge. This reflexive capacity has brought about some well-know=
n, important advances in our understanding of the scope and limitations of =
mathematical knowledge, such as are unknown in any other scientific field. =
Over the first half of the present century, important achievements were gai=
ned in this direction, in particular in those disciplines usually grouped u=
nder the common heading of metamathematics. These achievements (G?del's the=
orems is the classical example that immediately comes to mind) were certain=
ly limited to very specific domains, and did not necessarily have any signi=
ficant, direct impact on the work of the overwhelming majority of mathemati=
cians engaged in different areas of research. Still, they helped reinforcin=
g a point of view according to which any claims -be they historical, philos=
ophical and methodological- about the discipline of mathematics become mean=
ingful and worth of attention only insofar as they may be endorsed by forma=
l mathematical arguments. I call this latter point of view "the reflexive i=
mage of mathematics."=20
Echoes of the reflexive image of mathematics can be found in much philosoph=
ical work published in the twentieth century, as reflected in the absolute =
dominance of the foundationalist currents over this period of time (see bel=
ow p. 44). Normal mathematical research, on the other hand, was probably no=
t much affected by conceptions of this kind, but, as with other similar ide=
as, mathematicians have often resorted to the reflexive image when the need=
 arises to explain to themselves, or to others, the nature of their own bus=
iness. Bourbaki's attempts to define a formal theory of structures can, to =
a considerable extent, be understood in these terms: as an attempt to refle=
xively elucidate the notion of a "mathematical structure" and the significa=
nce of conceiving mathematics in terms of it. In fact, developing a reflexi=
ve, formal-axiomatic, elucidation of the idea of mathematical structure cou=
ld have proved useful not only as a general frame of reference, but also as=
 a tool for addressing some very specific, central open questions that Bour=
baki's adoption of the axiomatic and the structural images of mathematics m=
ade patent. One such central issue was the issue of selection.=20
=09Theory-selection and problem-choice are central questions in science in =
general, at the level of the images of knowledge. What individual scientist=
s select as their discipline of research, and the particular problems they =
choose to deal with in that particular discipline largely determines, or at=
 least conditions, the scope and potentialities of their own personal resea=
rch. What a community of scientists establishes as main open problems and m=
ain active sub-disciplines substantially influences the future development =
of that discipline as a whole. Clearly, the contents of the body of knowled=
ge directly delimit the potential selections of scientists. But on the othe=
r hand, these contents alone cannot provide clear-cut answers to the issue =
of selection. Criteria of selection are open to debate and, obviously, ther=
e are several possible factors that determine a particular scientist's choi=
ce, when confronted with a given body of knowledge.
=09Bourbaki was clearly conscious of the centrality of the issue of selecti=
on and, from the very beginning of the group's activities, considerable eff=
ort was invested in debating it. In the early meetings, that eventually led=
 to the creation of the core Bourbaki group, an important criterion for the=
 selection of issues to be treated in the projected treatise on analysis wa=
s their external applicability and their usefulness for physicists and engi=
neers. Over the first years of activities, however, given the more abstract=
 inclinations of certain members and the way in which the writing of the ch=
apters evolved, gradual changes affected the criteria of selection guiding =
the group's work.26=20
=09As the axiomatic approach gradually became a dominant concern for Bourba=
ki, the problem of selecting, and especially that of justifying, the most i=
nteresting theories to be included in the treatise increasingly became a pr=
essing one. As Henri Cartan wrote in retrospect, on the face of it the choi=
ce of axioms could seem to be completely arbitrary; in practice, however, a=
 very limited number of such systems constitute active mathematical researc=
h disciplines, since theories "built upon different axiomatic systems have =
varying degrees of interest" (Cartan 1958 [1980], 177). Dieudonn? did not h=
esitate to use the term "axiomatic trash" (1982a, 620), to designate theori=
es based on the axiomatic treatment of systems that he considered unimporta=
nt or uninteresting. But, what is actually the criterion for winnowing the =
chaff of "axiomatic trash" from the wheat of the mathematically significant=
 axiomatic systems?=20
=09Under the spell of the reflexive image of mathematics, it would be natur=
al, or at least plausible, to expect that an answer to the above-posed ques=
tion be given by means of a reflexive mathematical theory. The correct choi=
ces could, in this case, be endorsed by results attained work within a stan=
dard mathematical theory. Bourbaki's formulation of the theory of structure=
s could be seen as a possible response to that expectation. But on the othe=
r hand, considering the intellectual inclinations of the mathematicians inv=
olved in the Bourbaki project, one is justified in thinking that each membe=
r of the group had strongly conceived opinions of what should be considered=
 as mathematically interesting and what should not, independently of the el=
aboration of a formal theory of structures.27=20
=09At any rate, one can see how the thorough adoption of the axiomatic appr=
oach as the main tool for the exposition of mathematical theories, together=
 with the images of knowledge associated with that approach, create a direc=
t connection between the issue of selection and Bourbaki's formulation of t=
he theory of structures. As it happened, however, this theory did not effec=
tively provide answers to this, or to any other reflexive issue. Neverthele=
ss, Bourbaki's images of mathematics, and in particular the group's actual =
choices proved to be enormously fruitful in certain quarters of mathematics=
. Still more interesting, Bourbaki's criteria of selection have very often =
been accepted as if they were actually backed by such a reflexive theory, a=
nd the writings of some of the members of the group, particularly of Dieudo=
nn?, have strongly contributed to enhance this belief. This has contributed=
 to present Bourbaki's images of mathematics as fully backed by knowledge d=
rawn from the body of mathematics, and it is in this sense that the eternal=
 character usually attributed to the body of knowledge has come to be exten=
ded, in the picture of mathematics promoted by Bourbaki, to the images of k=
nowledge as well.
=09Perhaps the most interesting example of how this extension has worked co=
ncerns the issue of the "mother-structures", allegedly one of the central p=
illars associated with Bourbaki's mathematics. The role of the "mother stru=
ctures" appears in Bourbaki's Architecture manifesto as follows:=20
At the center of our universe are found the great types of structures, ... =
they might be called the mother structures ... Beyond this first nucleus, a=
ppear the structures which might be called multiple structures. They involv=
e two or more of the great mother-structures not in simple juxtaposition (w=
hich would not produce anything new) but combined organically by one or mor=
e axioms which set up a connection between them... Farther along we come fi=
nally to the theories properly called particular. In these the elements of =
the sets under consideration, which in the general structures have remained=
 entirely indeterminate, obtain a more definitely characterized individuali=
ty. (Bourbaki 1950, 228-29)
This is the heart of Bourbaki's conception of mathematics as a hierarchy of=
 structures, and it has been quoted and repeated very often. But, the inter=
esting point is, that this picture has nothing to do with Bourbaki's theory=
 of structures! The classification of structures according to this scheme i=
s mentioned several times in Bourbaki's volume on set theory, but only as a=
n illustration appearing in scattered examples. Many assertions that were s=
uggested either explicitly or implicitly by Bourbaki or by its individual m=
embers -i.e., that all of mathematical research can be understood as resear=
ch on structures, that there are mother structures bearing a special signif=
icance for mathematics, that there are exactly three, and that these three =
mother structures are precisely the algebraic-, order- and topological-stru=
ctures (or structures)- all this is by no means a logical consequence of th=
e axioms defining a structure. The notion of mother structures and the pict=
ure of mathematics as a hierarchy of structures are not results obtained wi=
thin a mathematical theory of any kind. Rather, they belong strictly to Bou=
rbaki's non-formal images of mathematics; they appear in non-technical, pop=
ular, articles, such as in the above quoted passage, or in the myth that ar=
ose around Bourbaki. And yet, because of the blurred mixing of the two term=
s, structures and "structures" in Bourbaki's work, they have been accorded =
a status of truth similar to the one accorded to other mathematical results=
 appearing in Bourbaki's treatise, namely, that of eternal truths.
Closely related to this issue is the relationship between Bourbaki's work a=
nd the development of the theory of categories. This theory, first formulat=
ed in 1942 by Samuel Eilenberg and Saunders Mac Lane and more vigorously el=
aborated since the 1960s by a steadily growing community of practitioners, =
provided a viable alternative to what the theory of structures promised: a =
general framework within which the various mathematical domains and the int=
errelations among them could be mathematically studied. It would be far bey=
ond the scope of the present article to discuss this issue in detail.28 Wha=
t is of direct relevance to the present discussion is that the very existen=
ce of such an alternative posed an interesting challenge to the picture pre=
sented in the El?ments and certainly to the underlying claim of eternal val=
idity for Bourbaki's structural (structural?) image of mathematics. As it h=
appened, category theory came to provide a useful language that was fruitfu=
lly used in many different mathematical domains but, very much like the the=
ory of structures, it attained rather little significance as an overall, or=
ganizational scheme for mathematical knowledge.=20

Another interesting example of the sweeping validity claims associated with=
 Bourbaki's work appears in a book published by Dieudonn? in 1977 (and tran=
slated into English in 1982) under the name of A Panorama of Pure Mathemati=
cs. As seen by Nicolas Bourbaki. Dieudonn? presents in this book an overvie=
w of many branches of mathematics and of the main problems addressed in eac=
h of them. He put forward a picture of mathematics as divided into two grea=
t parts: a "classical" and a "live" one. The classical part is that part of=
 mathematics embodied in the various volumes of Bourbaki's El?ments. No mor=
e and no less. The live part is that one which is still in a process of bei=
ng constituted: it is still changing and therefore it is unstable, but even=
tually, as it stabilizes, it will be added to the El?ments and thus it will=
 become part and parcel of "classical" mathematics. Moreover, so asserts Di=
eudonn?, the reader interested in knowing what are the most important parts=
 of this live mathematics, will find it by consulting the proceedings of th=
e Seminar Bourbaki, currently held a the University of Paris. This is obvio=
usly a very "non-classical" definition, one may say, of what classical math=
ematics is, and also of what live mathematics is. Not only did Dieudonn? en=
dorse his classification and his whole selection with his professional auth=
ority, which was rather well established by then, but he also added the sug=
gestion that this way of presenting the core of mathematics, unlike earlier=
 ones, is one that will remain unchanged from now on!. In other words: Bour=
baki's picture of mathematics does not only include a body of mathematics c=
omposed of eternal truths, but this is also the case for the concomitant im=
ages of mathematics. At any rate, one has to take into account that by the =
time this book was published, Bourbaki was well past its heyday and thus Di=
eudonn? was perhaps trying to reinforce again a point that may have already=
 been less obvious that it was in the recent past.
=09Yet a third, and last, instance I would like to mention here, of Bourbak=
i's way to present a certain system of images of knowledge as eternally val=
id, is Bourbaki's historiography. Bourbakian historiography is manifest mai=
nly in the collection of articles published as El?ments d'histoire des math=
?matiques (1969), as well as through the many historical writings of indivi=
dual Bourbaki members, especially Dieudonn?. This historiography has receiv=
ed considerable attention and criticism,29 and this is not the place to dis=
cuss it in detail, except for what pertains the point at issue here. Bourba=
ki's historiography is the classical example of that approach according to =
which the importance of mathematical ideas in the past is judged by ponderi=
ng their relevance to present conceptions. Bourbaki is not the only represe=
ntative of this trend in the history of mathematics, but what is singular a=
bout Bourbaki is that the framework of reference he adopted for making hist=
orical judgment is one that, as was said above, is meant to remain unchange=
d in the future as well! Thus, Bourbaki (especially Dieudonn?) interestingl=
y combine in historiography two different aspects: a Whigish approach to pa=
st history and a belief in the eternal character, not only of the truth of =
mathematical theorems and results, but also of its present organization.
=09An interesting example of the way in which the idea of structure and the=
 importance accorded to it in twentieth century mathematics enters Dieudonn=
?'s historiography as a criterion for retroactive historical judgment is ma=
nifest in his account of the history of algebraic geometry (Dieudonn? 1985)=
. This account distinguishes seven different periods in the development of =
the discipline. The first four periods, from 400 B.C. to about 1866, up to =
and including the works of Riemann and Abel, cover the first twenty-six pag=
es of his book. The fifth period ("Development and Chaos": 1866-1920) is di=
scussed in the next thirty-two pages. Most of the discussion in this chapte=
r (pp. 29-35) is devoted to a classical article by Richard Dedekind and Hei=
nrich Weber (1882), and this is -as Dieudonn? explained- because of its pro=
ximity to modern, structural ideas. As Dieudonn? himself writes, however, f=
or this very reason the article had a limited impact on contemporary mathem=
aticians. The sixth period, of only thirty years, is the one to which a mos=
t detailed analysis is accorded: "New Structures in Algebraic Geometry (192=
0-1950)". What characterizes this period, in Dieudonn?'s view, is the fact =
that "at the beginning of the twentieth century the general idea of the str=
uctures underlying diverse mathematical theories became completely consciou=
s" (p. 59).=20
=09But Dieudonn?'s criteria for historical research, based on choices simil=
ar to those that led his own mathematical research, is not left implicit in=
 the different attention accorded to the various historical stages of his s=
tory. Dieudonn? stated his historiographical approach explicitly when he wr=
ote that the "algebraic school", although chronologically last, would be tr=
eated first because "in the light of future history, it is the algebraic in=
clination that exercised the most profound influence." Moreover, the altern=
ative approaches to it were at their time just contributing to chaos; they =
were "attracted to one aspect or another of Riemann's works, and thus are b=
orn several schools of algebraic geometry that tend to diverge up to the th=
reshold of mutual incomprehension" (p. 27).

In spite of the fact that Bourbaki's theory of structures does not help sol=
ving any of the open questions at the level of the images of knowledge, Bou=
rbaki has nevertheless been fond of presenting his own choices as if they w=
ere fully justified on purely mathematical grounds, as definite and final, =
and as unbiased by personal or socially conditioned tastes. Moreover, Bourb=
aki's selection allegedly does not imply a value judgment. For example, gro=
up theory, despite its acknowledged importance, is not included in Bourbaki=
's treatise because "we cannot say that we have a general method of attack"=
, and therefore the systematic presentation attained in the other theories =
developed in the treatise cannot be introduced for this one. But the questi=
on still remains open: how does the group justify its own choices?
=09This question brings us to an additional, important, related issue. Ther=
e is usually a high degree of agreement among mathematicians as to what sho=
uld legitimately be considered to be part of the body of mathematical knowl=
edge: knowledge has to be endorsed by some kind of proof. The absence of de=
bate characteristic in general of the body of mathematics is often taken, b=
y analogy, to be the desired state of affairs at the level of the images of=
 knowledge as well. From reading texts like the El?ments, for instance, one=
 might tend to think that there are not, and there should be no debates at =
all at the level of the images of mathematics. If, nevertheless, debate doe=
s eventually arise regarding the images of knowledge, then it is often deal=
t with in one of several ways; either =20
1. a mathematical theory is proposed within which the debate may be safely =
decided by means of proof, or
2. it is resolved by resort to authority, or alternatively,
3. it is simply ignored.
Had Bourbaki's theory of structures had something substantive to say about =
the hierarchy of structures and related issues, then it would have provided=
 an alternative of type (1) above, but clearly it failed to do so. Lacking =
a reflexive argument such as the theory of structures could have provided, =
Bourbaki has resorted to the second best alternative to resolve the issue o=
f selection: authority.=20
 =09Like in other aspects of Bourbaki's work, Dieudonn? has taken the lead =
in expressing his opinion on this issue. Although he has also advanced some=
 substantive arguments, authority frequently seems to be his soundest justi=
fication for Bourbaki's choice. Thus he has written:
No one can understand or criticize the choices made by Bourbaki unless he h=
as a solid and extended background in many mathematical theories, both clas=
sical and more recent. (Dieudonn? 1982a, p. 623)
Sheer authority, however, confers a taste of arbitrariness to this claim. S=
ince it is generally considered unacceptable to accord the arbitrary an imp=
ortant role in mathematics, some additional claims have been advanced as fu=
rther justification for Bourbaki's choice-criteria. The following quotation=
 is but one example where Dieudonn? presents what he sees, in retrospect, a=
s Bourbaki's selection criteria:=20
=CCIn spite of its initial aim at universality, the scope of the Bourbaki t=
reatise has finally been greatly reduced (although to a still respectable s=
ize) by successive elimination of:
1. the end products of theories, which do not constitute new tools;
2. the unmontivated abstract developments scorned by the great mathematicia=
ns;
3. important theories (in the opinion of great mathematicians) that are far=
 from clear descriptions in terms of an interplay of perspicuous  structure=
s; examples are finite groups or the analytic theory of numbers. (Dieudonn?=
 1982a, 620)
Like any other list of criteria meant to provide a useful guide for choice =
in either mathematical or historical research, this one can be criticized o=
n many grounds. In particular, it is interesting to notice that these crite=
ria are nothing but a reformualtion of the criteria of professional authori=
ty. But the really interesting issue that these criteria raise, from the po=
int of view of the present article, is that rather than solving it, they on=
ly underscore the problem involved in establishing once and for all, as Bou=
rbaki wanted, a comprehensive system of images of mathematical knowledge, t=
hat would attain the status of eternal truth usually accorded to results be=
longing to the body of knowledge.

To summarize, then, one can see how Bourbaki's images of mathematics, espec=
ially as formulated and promoted by the group's most active speaker, Jean D=
ieudonn?, put forward the idea that not only the body of mathematical knowl=
edge is a collection of eternal truths, but that this is also the case for =
the images of knowledge as well. In particular, the structural image of mat=
hematics is the ultimate stage of a necessary process of historical develop=
ment, and it is bound to remain unchanged in the future. After the publicat=
ion of the El?ments, so suggested Dieudonn?, future developments in mathema=
tics would proceed squarely within the basic framework stipulated by the st=
ructural image put forward in this treatise: more complex structures would =
perhaps be developed, that combine in new ways the mother-structures and th=
e other, already known, structures built on the latter. Above all, new and =
more sophisticated knowledge would be gained about all these structures. Bu=
t beyond that, our knowledge that mathematics is a hierarchy of structures =
and that mathematical knowledge advances by further elucidating the individ=
ual structures that compose the hierarchy - this image would remain unchang=
ed and as eternal as any of the specific theorems that have been proved abo=
ut any of those individual structures.


4. Some Recent Debates
>From the 1950s to the late 1970s, Bourbaki's images of knowledge exerted a =
tremendous influence on mathematical research and teaching, especially in t=
he "pure" branches, all over the world. This influence surely counts as one=
 of the main factors behind the apparently robust status of the belief in t=
he eternal character of mathematical knowledge over this period of time. Th=
is belief is far from having disappeared from the mathematical scene (and p=
erhaps this is not without justification), but at the same time, interestin=
g debates have arisen around it. In the present section, I mention some of =
the most recent, dissenting views.
In 1980, the mathematician Morris Kline published a provocative book entitl=
ed Mathematics. The Loss of Certainty (Kline 1980). The main thesis of the =
book is that the received view of mathematics, according to which this disc=
ipline "is regarded as the acme of exact reasoning, a body of truths in its=
elf, and the truth about the design of nature," is plainly false! Kline pre=
sented in his book a historical account of the rise of mathematics to the u=
nparalleled heights of "prestige, respect and glory" that were accorded to =
it from ancient times and well into the nineteenth century. This developmen=
t, however, was followed by what Kline sees as a total debacle in which all=
 certainty about the truth of mathematics and, especially, concerning the q=
uestion of which approach to the foundations of mathematics is correct and =
secure, was lost. In Kline's words:
It is now apparent that the concept of a universally accepted, infallible b=
ody of reasoning -the majestic mathematics of 1800 and the pride of man- is=
 a grand illusion. Uncertainty and doubt concerning the future of mathemati=
cs have replaced the certainties and complacency of the past. The disagreem=
ent about the foundations of the "most certain" science are both surprising=
 and, to put it mildly, disconcerting. The present state of mathematics is =
a mockery of the hitherto deep-rooted and widely reputed truth and logical =
perfection of mathematics. (Kline 1980, 6)
"The Age of Reason", Kline concluded in the introductory chapter of his boo=
k, "is gone."
=09The details of Kline's arguments and the question whether or not they le=
ad to his sweeping, appalling, conclusions will not concern us here. The in=
terested reader can consult the book and judge this by herself. There is no=
 doubt, however, that Kline's claims, whether well taken or not, whether su=
pported by sound historical evidence or not, were rather uncommon at the ti=
me of their publication, especially coming from a prominent mathematician l=
ike himself.
=09Kline's book did not immediately give raise to any kind of open controve=
rsy. If one has to judge according to published reactions, then the conclus=
ion is that the book was largely ignored by mathematicians. The few publish=
ed reviews of this book suggest that the mathematical community may have ev=
en been hostile to the kind of arguments put forward by Kline.30 At the sam=
e time, however, one can see in retrospect that Kline pointed to a directio=
n that was soon to be followed by others, who would undertake a reexaminati=
on of the character of eternal truth commonly attributed to mathematical kn=
owledge.
=09Substantial evidence that such a reexamination was under way appeared in=
 1985 in a collection of articles edited by Thomas Tymoczko, under the name=
 New Directions in the Philosophy of Mathematics. The articles in this coll=
ection, written by mathematicians, as well as by philosophers and historian=
s, put forward a philosophy of mathematics that Tymoczko calls "quasi-empir=
icist", and that is opposed to the view that had dominated discourse in thi=
s domain, at least since the 1920s. Tymoczko called this formerly dominant =
view "foundationalism"[EC1], and he characterized it as the search for the =
true foundations of what is assumed, in the first place, to be a system of =
certain, unchanging, knowledge. This view includes, of course, three main s=
chools of philosophy of mathematics in the present century, namely, logicis=
m, formalism and intuitionism.=20
=09The quasi-empiricist works that Tymoczko collected in his volume share a=
 common interest in the processes of production, communication and change o=
f mathematical knowledge, rather than focusing on the finished, and alleged=
ly definitive, versions of it. Also, they coincide in stressing that the na=
ture of mathematics can only be elucidated when this science is considered =
to be an organic, lively body of knowledge, and that the analysis of its fo=
undations is only a very partial perspective of this more general task. The=
y stressed that mathematical knowledge arises as part of a social process i=
n which elements of uncertainty, such as plain mistakes, empirical consider=
ations, heuristic factors, and even tastes and fads, may play some role. Th=
is view does not necessarily imply a relativistic account of mathematics, b=
ut it does dispute, perhaps from a perspective somewhat different from the =
one suggested by Kline, some basic, accepted beliefs concerning the nature =
of mathematical knowledge as a body of eternal, unshakable truth.31
=09Tymoczko's collection included recent articles, as well as less recent o=
nes, such as those by Imre Lakatos, who began publishing his idiosyncratic =
work on the philosophy of mathematics in the late sixties. But one main, di=
rect, motivation behind Tymoczko's publication was the recent rise to promi=
nence of new kinds of proofs that departed from the classical Euclidean par=
adigm, a long-dominant one, on which the classical view of the eternal natu=
re of mathematical truth was based. Among those new kinds of proofs, especi=
al attention was accorded to "computer-assisted proofs", (e.g. to the four =
color problem), but also to "very long proofs" (e.g. to the simple, finite =
groups classification theorem), and to proofs that established that a theor=
em was true with an "extremely high probability", rather than with absolute=
, Euclidean or deductive, certainty (e.g. Rabin 1976 on the distribution of=
 prime numbers). This is not the place to describe all these kinds of proof=
s and the philosophical questions they raise.32 The point here is simply to=
 make clear that some actual mathematical developments raised pressing ques=
tions and posed new challenges that somehow clashed with the received conce=
ption of mathematics as a body of eternal truths. These questions were addr=
essed mainly by philosophers and historians of mathematics, and under the i=
nfluence of their works some observers went so far as to pronounce the clas=
sical conception of proof officially dead (Horgan 1993). The reactions of m=
ost working mathematicians to these developments were at this stage either =
indifferent or hostile to the conclusions that some non-mathematicians were=
 deriving from their second-hand knowledge of them (Thurston 1994).
=09A noteworthy event in the debate on the eternal character of mathematica=
l truths took place quite recently, when some mathematicians -in fact, some=
 very prominent mathematicians- came forward with their own proposals to ch=
ange the accepted canons of mathematical publishing. By doing so, they anti=
cipated that a broader spectrum of what constitutes the actual process of m=
athematical research and knowledge will become public and will be shared by=
 the mathematical community at large. This process will affect the concepti=
on of mathematics as a body of eternal truths, and it will contribute, so t=
hese mathematicians hope, to the enhanced development of their discipline.
=09A by now well-known manifestation of this trend was an interchange publi=
shed over the pages of the Bulletin of the American Mathematical Society in=
 1993 and 1994. It started with an article by Arthur Jaffe and Frank Quinn,=
 both distinguished mathematicians. As a matter of fact, Jaffe, a mathemati=
cal physicist from Harvard, is presently President of the AMS. Their articl=
e bears the title "Theoretical Mathematics: Toward a Cultural Synthesis of =
Mathematics and Theoretical Physics". According to the authors, recent even=
ts in the development of mathematics and physics dictate the need for a red=
efinition of the relations between the two sciences. In particular, they cl=
aimed, there has recently been an intense activity in physics that has yiel=
ded many new insights into pure mathematical fields. Some of these results =
were eventually taken over by mathematicians and reworked according to thei=
r professional tastes, but originally they were produced by the physicists =
without themselves abiding by the standards set by the mathematical communi=
ty for their own works. Jaffe and Quinn had in mind, among others, the rece=
nt work of Edward Witten in string theory, and they thought that mathematic=
ians should encourage the production of works similar to this one. In their=
 view, without an active initiative to do so, the current professional more=
s would hinder such contributions and thus cut a vital source for inspirati=
on and insight for mathematics.=20
=09Faced with such a situation, the article suggests the need to adopt in m=
athematics a division of labor accepted in physics throughout the present c=
entury, namely, that between theoretical and experimental physics. How is t=
his division translatable into mathematics? Jaffe and Quinn compared the in=
itial stages of mathematical discovery, involving speculation, intuition an=
d convention, to the work of the theoretical physicists. Like experiment in=
 physics, rigorous mathematical proof is introduced only later in order to =
correct, refine and validate the results and insights obtained in the earli=
er stage. Thus, while admitting that the terms "theoretical" and "experimen=
tal" mathematics may be somewhat confusing at first, Jaffe and Quinn sugges=
ted the following prescription for a healthy, future development of mathema=
tics:
The mathematical community has evolved strict standards of proof and norms =
that discourage speculation. These are protective mechanisms that guard aga=
inst the more destructive consequences of speculation; they embody the coll=
ective mathematical experience that the disadvantages outweigh the advantag=
es. On the other hand, we have seen that speculation, if properly undertake=
n, can be profoundly bebeficial. Perhaps a more conscious and controlled ap=
proach that would also allow us to reap the benefits but avoid the dangers =
is possible. The need to find a constructive response to the new influences=
 from theoretical physics presents us with both an important test case and =
an opportunity.
Mathematicians should be more receptive to theoretical material but with sa=
feguards and a strict honesty. The safeguards we propose are not new; they =
are essentially the traditional practices associated with conjectures. Howe=
ver a better appreciation of their function and significance is necessary, =
and they should be applied more widely and more uniformly. Collectively, ou=
r proposals could be regarded as measures to ensure 'truth in advertising,'=
 [e.g., :] "Theoretical work should be explicitly acknowledged as theoretic=
al and incomplete; in particular, a major share of the credit for the final=
 result must be reserved for the rigorous work that validates it." (Jaffe a=
nd Quinn 1993, 10)
=09It was clear to the editors of the Bulletin that a proposal of this kind=
 would not pass in silence. Even before publication they asked several lead=
ing mathematicians to write their opinions and reactions, to be published i=
n a forthcoming issue of the journal (Atiyah et al. 1994; Jaffe & Quinn 199=
4). The published reactions ranged from a total rejection (e.g., by Saunder=
s Mac Lane), to a criticism of the general, authoritative, tone adopted by =
the authors when suggesting new standards for publication (e.g., by Michael=
 Atiyah, Armand Borel, and Benoit Mandelbrot), to a general agreement in pr=
inciple but disagreement in the details of the proposal (by William Thursto=
n and Albert Schwartz), to a disagreement in principle but agreement with s=
ome of the details (by Ren? Thom).33
=09But the debate remained open and one may expect, if only for the promine=
nce of the mathematicians involved, that the issues raised by it will not b=
e forgotten very soon. As a matter if fact, on February 12, 1996, a colloqu=
ium was held at Boston University, on "Proof and Progress in Mathematics", =
which was basically a follow up of this interchange. Jaffe and Mac Lane wer=
e again among the discussing parties, together with other mathematicians, s=
uch as Gian-Carlo Rota, from MIT, and the Harvard mathematician Barry Mazur=
. New issues were raised in this meeting, which in retrospect seem inevitab=
le. Such is the case, e.g., of the role of electronic communications among =
mathematicians, Internet, etc. The pervasiveness of these new media raises =
the need to redefine some well-established concepts pertaining the mathemat=
ical profession: publishing, definitive versions, authorship of ideas and r=
esults, etc.34=20

Parallel to the Jaffe-Quinn proposal for reconsidering the accepted norms o=
f publication in mathematics, the role of rigor in proof and -implicitly at=
 the very least- of the eternal character of mathematical truth, I want to =
mention here an additional, similar, debate involving prominent mathematici=
ans. This one was sparkled by Doron Zeilberger, from Temple University, in =
an article bearing the provocative name of "Theorems for a Price: Tomorrow'=
s Semi-Rigorous Mathematical Culture." Based, among others, on the innovati=
ons implied by his own important mathematical contributions, Zeilberger att=
empted in this article to attack a conception of mathematics, which, althou=
gh still dominant today, is in his view actually obsolete and bound to be c=
hanged by a new mathematical culture. Today's conception was characterized =
by Zeilberger as follows:
The most fundamental precept of the mathematical faith is thou shalt prove =
everything rigorously. While the practitioners of mathematics differ on the=
ir views of what constitutes a rigorous proof, and there are fundamentalist=
s who insist on even a more rigorous rigor than the one practiced by the ma=
instream, the belief in this principle could be taken as the defining prope=
rty of mathematician. (Zeilberger 1994, 11. Italics in the original)
=09This conception, promised Zeilberger, will soon be preserved only by a s=
mall sect of fringe mathematicians, that, in spite of the deep changes expe=
cted, will choose to keep abiding by the now orthodox conception. In order =
to support his claim and make explicit whom he refers to with this descript=
ion, he cites the 1993 article by Jaffe and Quinn. The reader thus understa=
nds that Zeilberger is going to present a truly radical proposal for the fu=
ture of mathematics.=20
=09Zeilberger makes his point by referring to the so-called algorithmic pro=
of theory of hypergeometric identities, a field to which he made significan=
t contributions. This theory considers identities involving certain functio=
ns, and proves or refutes them by means of algorithms that reduce any given=
 identity of this kind to an auxiliary one, involving only specific polynom=
ials. Today the theory can be successfully applied to a wide range of known=
 identities, but, as Zeilberger explains, it is natural to expect that in t=
he future one might construct examples of identities, whose reduction using=
 the known algorithms in any computer will involve prohibitive amounts of r=
unning time or of memory. Performing the algorithms in this case would lead=
 to absolute certainty concerning the truth or falsity of the identities, b=
ut the price (in dollars) one would have to pay for doing so would be enorm=
ous. On the other hand, it is possible to apply a different kind of algorit=
hms from which we will be able to answer the same question, not with full c=
ertainty, but with a very high probability and for free, or for a very low =
price in terms of computer resources.
=09I already mentioned above "probabilistic" proofs, namely, arguments that=
 assign a very high probability to statements of the kind "Theorem X is tru=
e." Michael Rabin had published one such argument in 1976 concerning the st=
atement that a certain number is prime. Rabin devised an algorithm, each it=
eration of which raises the probability in question. Thus, the idea of a pr=
obabilistic proof is not a new one. Still, Rabin did never claim that this =
should become a mainstream way of supporting mathematical truth. Moreover, =
it seems quite clear that Rabin would be very much pleased to have a deduct=
ive arguments to prove, in the classical and (by implication) conclusive wa=
y, what his probabilistic proof seemed only to support with a very high lik=
elihood. Zeilberger, on the contrary, is explicitly arguing for the adoptio=
n of these kinds of proofs as the standard, mainstream vindication of the t=
ruth of a mathematical statement. Zeilberger invokes two arguments to suppo=
rt his position: First, he says, it is likely that few new, non-trivial, re=
sults might be proved through deductive arguments. Second: the price of the=
 latter will become increasingly high. It is pertinent to quote here Zeilbe=
rger himself:
As wider classes of identities, and perhaps even other kinds of classes of =
theorems, become routinely provable, we might witness many results for whic=
h we would know how to find a proof (or refutation); but we would be unable=
 or unwilling to pay for finding such proofs, since "almost certainty" can =
be bought so much cheaper. I can envision an abstract of a paper, c. 2100, =
that reads, "We show in a certain precise sense that the Goldbach conjectur=
e is true with probability larger than 0.99999 and that its complete truth =
could be determined with a budget of $ 10 billion." (...)
As absolute truth becomes more and more expensive, we would sooner or later=
 come to grips with the fact that few non-trivial results could be known wi=
th old-fashioned certainty. Most likely we will wind up abandoning the task=
 of keeping track of price altogether and complete the metamorphosis to non=
rigorous mathematics. (Zeilberger 1994, 14)
=09A reply to Zeilberger's article was published very soon by George E. And=
rews. Andrews is himself a mathematician of no less merits than Zeilberger =
(in fact the two have collaborated on many occasions). It is instructive to=
 read Andrews in order to realize that, in spite of the strong arguments pu=
t forward, and in spite the verve expressed, by Zeilberger, Jaffe, Quinn, a=
nd others, the idea of eternal truth in mathematics will not disappear so s=
oon, if only for reasons that touch to the sociology of the profession, but=
 certainly also for reasons deeper than that. In a formulation that might w=
armly be adopted by many colleagues Andrews disputed Zeilberger's position =
with the following words:
Through the summer of 1993 I was desperately clinging to the belief that ma=
thematics was immune from the giddy relativism that has pretty well destroy=
ed a number of disciplines in the university. Then came the October Scienti=
fic American and John Horgan's article, "The death of proof" [Horgan 1993].=
 The theme of this article is that computers have changed the world of math=
ematics forever, in the process making proof an anachronism. Oh well, all m=
y friends said, Horgan is a non-mathematician who got in way over his head.=
 Apart from his irritating comments and obvious slanting of the material, "=
The death of proof" actually contains interesting descriptions of a number =
of important mathematics projects. Indeed, as W. Thurston has said [Thursto=
n 1994] "A more appropriate title would have been 'The Life of Proof'."=20
Then came [Zeilberger's article] (...) Unlike Horgan, Zeilberger is a first=
-rate mathematician. Thus one expects that his futurology is based on firm =
ground. So what is his evidence for this paradigm shift? It was at this poi=
nt that my irritation turned to horror. (Andrews 1994, 16)
Andrews described in some technical detail why, in his view, Zeilberger spe=
cific arguments do not support his claim. At the same time Andrews stressed=
 an important component of mathematical knowledge that in his view Zeilberg=
er's perspective failed to stress: the role of insight.
Perhaps it is also relevant to cite here a further reaction to Zeilberger's=
 and Andrews's articles, that appeared under the title of "Making Sense of =
Experimental Mathematics" (Borwein et al. 1996). This article attempts to p=
ut the whole debate raised by mathematicians such as Jaffe, Quinn and Zeilb=
erger, in a broader context, and to find a common ground that might be acce=
pted by a larger portion of the mathematical community. The article put for=
ward some arguments which are interesting in themselves, but that is not th=
e point I want to stress here. What I find worth of special attention is th=
e fact that one of its authors, Jonathan Borwein, works at the "Center for =
Experimental and Constructive Mathematics", Simon Fraser University. This i=
s only one of this kind of institutions active today in many universities a=
round the world. Thus, while the debate on new ways to legitimize mathemati=
cal truth is still an open one, new institutions are being built which alre=
ady promote work based on the new principles. One should not be surprised, =
then, to realize that Zeilberger's vision of "theorems for a price" might b=
ecome reality, much sooner than he has envisaged, though not literally in t=
he sense described in his article: it is not unlikely that financial suppor=
t for "Centers for Experimental Mathematics" around the world might soon su=
rpass the one allocated for more traditionally-oriented departments, thus d=
ictating, for a price, what kinds of theorems are going to be proved and in=
 which direction mathematics is going to advance in the foreseeable future.=
 Institutional factors have been decisive throughout history in shaping the=
 course of development of mathematical ideas (through education, grants, ap=
pointments, promotion, etc.), and given the present state of academic resea=
rch, such consideration will only become increasingly important.
=09At any rate, it is not the aim of this article to elucidate the future c=
ourse of mathematical research into one of the directions suggested by the =
mathematicians mentioned in the foregoing pages. The aim of this section is=
 just to indicate an interesting turn that the idea of eternal truth in mat=
hematics has undergone over the last ten years. This makes more perspicuous=
 the relevance of the historical analysis that was presented in the precedi=
ng sections. If this article had been written in the early eighties it coul=
d have started with the following words: "Mathematics is the scientific dis=
cipline in which the idea of eternal truths is most deeply entrenched. In f=
act, unlike other sciences, twentieth-century developments have only streng=
thened this historically conditioned tendency." However, in view of the dev=
elopment mentioned, I must erase the second sentence of the quotation, and =
instead start as follows: "Mathematics is the scientific discipline in whic=
h the idea of eternal truths has historically been most deeply entrenched, =
although recent developments have modified this to a certain degree, in a d=
irection whose actual significance is still to be definitely evaluated."=20

5. Summary and Concluding Remarks
The foregoing sections discussed the views of some leading twentieth-centur=
y mathematicians concerning the status of truth in their discipline. Neithe=
r Bourbaki's nor Hilbert's views in this context is monolithic, yet, in gen=
eral they share the belief in the eternal character of mathematical truth w=
hich has basically been unchallenged throughout history, and still remains =
so. The interesting debates and nuances that this issue raises pertain to t=
he ways of achieving these truths.=20
=09In Bourbaki's conception, the conjunction of the structural, the axiomat=
ic and the reflexive images of mathematical knowledge together produce an i=
mage of mathematics that, besides leading to the discovery of new eternal m=
athematical truths in a unprecedentedly effective way, bears itself the cha=
racter of eternal truth: Bourbaki's image of mathematics is bound to remain=
 unchanged as well as the truths to which discovery it leads.=20
=09Hilbert's views, on which Bourbaki's are supposedly based, were much mor=
e multifarious. The eternal truths of mathematics are, in his view, attaine=
d in complex ways. The axiomatic method was seen as a very useful, but in n=
o way infallible, tool leading to such truths. It helps mainly in the ident=
ification of precisely those places where falsity or contradiction has ente=
red into mathematical reasoning, but even a mindful and able use of the met=
hod leaves much room for error, uncertainty, innovation and need for change=
. In his published works on physics, for example, Hilbert's axiomatic treat=
ment of theories (e.g., radiation theory or general relativity) suggests an=
 air of definiteness, but in his lectures he put forward a somewhat more te=
ntative approach. And certainly, Hilbert did not think that the axiomatic m=
ethod, or any other mathematical idea for that matter, can lead to a defini=
tive scheme for organizing science.
=09My discussion of Hilbert, Bourbaki, and eternal truths in mathematics, a=
lso helps clarifying, I believe, the background to the recent debates menti=
oned in =A7 4 above. These debates are clearly debates about the images of =
mathematical knowledge. They attempt to establish the disciplinary boundari=
es of legitimate mathematical knowledge. They do not in general question th=
e eternal character of the truths that currently exist in, and that must be=
 added to, the body of mathematical knowledge. Rather they question whether=
 or not, by accepting new forms of legitimation, new 'truths' are going to =
be accepted which perhaps will bear an essentially different, and therefore=
 undesired, character. On the one side of the debate are those who claim th=
at departing from the established model of proof, basically as embodied in =
Bourbaki's textbooks, will be detrimental for the future of mathematics as =
we know it today, precisely because it will cast serious doubt on the chara=
cter of the kind of truth involved in it. The other party involved in this =
debate does not seem to wish to change the character of mathematical truth =
as such. Rather, their claim is that introducing additional, legitimate mod=
els of proofs will not threat mathematics as a science of certain knowledge=
, and at the same time it will significantly enlarge its scope.
=09It therefore seems that Bourbaki's ambition of establishing once and for=
 all the images of mathematics according to which mathematical research wil=
l have to proceed in the future is being questioned today in directions, an=
d with an intensity, that not even Bourbaki's critics could have envisaged =
in the past.



Cohn Institute for History=20
and Philosophy of Science=09
Tel-Aviv University
Ramat Aviv 69978 ISRAEL
corry@post.tau.ac.il


References

Andrews, G.E., 1994. "The Death of Proof? Semi-Rigorous Mathematics? You've=
 got to be kidding!" Math. Int. 16 (4): 16-18.
Atiyah, M. et al., 1994. Responses to Jaffe and Quinn 1994. Bull AMS 30: 17=
8-207.
Beaulieu, L,. 1993. "A Parisian Caf? and Ten Proto-Bourbaki Meetings (1934-=
35)." Math. Int. 15, 27-35.
----. 1994. "Dispelling the Myth: Questions and Answers About Bourbaki's Ea=
rly Work, 1934-1944." In Ch. Sasaki et al. (eds.) The Intersection of Histo=
ry and Mathematics, Basel/Berlin/Boston, Birkh?user (1994), 241-252
Bernays, P.,  1922. "Die Bedeutung Hilberts f?r die Philosophie der Mathema=
tik." Die Naturwissenschaften 10: 93-99.
Born, M. 1922. "Hilbert und die Physik." Die Naturwissenschaften 10, 88-93.
Borwein, J. et al., 1996. "Making Sense of Experimental Mathematics." Math.=
 Int. 18 (4): 12-18.
Bourbaki, N., 1950. "The Architecture of Mathematics." Am. Math. Mo.  67, 2=
21-232.
----. 1969. El?ments d'histoire des math?matiques (2d. edition). Paris, Her=
mann.
Cartan, H., 1958 [1980]. "Nicolas Bourbaki and Contemporary Mathematics." M=
ath. Int. 2, 175-180. (English transl. by Kevin Lenzen of "Nicolas Bourbaki=
 und die heutige Mathematik." Arbeitsg. Forsch. Landes Nordheim 76 (1958), =
5-18.)
Corcoran, J., 1980. Review of Kline 1980. Math. Rev. 82e:03013.
Corry, L., 1989. "Linearity and Reflexivity in the Growth of Mathematical K=
nowledge." Science in Context 3, 409-440.
----. 1996. Modern Algebra and the Rise of Mathematical Structures. Boston,=
 Birkh?user (Science Network, Vol 17).
----. 1997. "David Hilbert and the Axiomatization of Physics (1894-1905)." =
Arch Hist. Ex. Sci.  51 (Forthcoming).
----. 1997a "Hermann Minkowksi and the Principle of Relativity." Arch. Hist=
. Ex. Sci. 51 (Forthcoming).
----. 1998 "David Hilbert and Physics", in  J.J. Gray (ed.) The Symbolic Un=
iverse: Geometry and Physics, 1890-1930, Oxford, Oxford University Press (F=
orthcoming).
Dedekind, R. and H. Weber, 1882. "Theorie der algebraischen Funktionen eine=
r Ver?nderlichen." Jour. r. ang. Math. 92, 181-290.
Detlefsen, M. and M. Luker, 1980. "The Four-Color Problem and Mathematical =
Proof." Journal of Philosophy 77: 803-24.
Dieudonn?, J. 1962. "Les m?thodes axiomatiques modernes et les fondements d=
es math?matiques." In F. Le Lionnais (ed.) Les grands Courants de la Pens?e=
 Math?matique (Second, enlarged edition), Paris, Blanchard, 443-555
----. 1982. A Panorama of Pure Mathematics. As seen by Nicolas Bourbaki, Ne=
w York, Academic Press. (English trans. by I.G. Mac Donald of Panorama des =
math?matiques pures. Le choix bourbachique (1977).)
----. 1982a. "The Work of Bourbaki in the Last Thirty Years", Notices AMS 2=
9, 618-623.
----. 1985. History of Algebraic Geometry. An Outline of the Historical Dev=
elopment of Algebraic Geometry, Monterrey, Wadsworth.
Fang, J., 1970. Bourbaki: Towards a Philosophy of Modern Mathematics I, New=
 York, Paideia Press.
Grattan-Guinness, I., 1979. Review of Dieudonn? (ed.) Abr?g? d'histoire des=
 math?matiques, 1700-1900, 2 vols., Paris, Hermann (1978). Annals of Scienc=
e 36, 653-655
Hertz, H., 1894 [1956]. The Principles of Mechanics Presented in a New Form=
, New York, Dover (English translation of Die Prinzipien der Mechanik in ne=
uem Zusammenhange dargestellt, Leipzig (1894).)
Hilbert, D., 1899. Grundlagen der Geometrie (Festschrift zur Feier der Enth=
?llung des Gauss-Weber-Denkmals in G?ttingen), Leipzig, Teubner.
----. 1902. "Mathematical Problems", Bulletin AMS 8, 437-479.
----. 1903. Grundlagen der Geometrie (2d, revised edition - with five suppl=
ements), Leipzig, Teubner.
----. 1916-17. Die Grundlagen der Physik, II, Ms. Vorlesung WS 1916-17, ann=
otated by R. B?r, Bibliothek des Mathematischen Seminars, Universit?t G?tti=
ngen.
----. 1918. "Axiomatisches Denken." Math. Ann. 78, 405-415.
----. 1922-3. "Wissen und mathematisches Denken." WS 1922/23, Ausgearbeitet=
 von W. Ackerman, Mathematisches Institut Universit?t G?ttingen (?berarbeit=
eter Nachdruck 1988).
----. 1919-20 [1992]. Natur und Mathematisches Erkennen: Vorlesungen, gehal=
ten 1919-1920 in G?ttingen. Nach der Ausarbeitung von Paul Bernays (Edited =
and with an English introduction by David E. Rowe, 1992), Basel, Birkh?user=
.
Horgan, J., 1993. "The Death of Proof." Scientific American 269 (4), 74-103=
.
Huntington, E., 1902. "Simplified Definition of a Group." Bulletin AMS 8, 2=
96-300
Jaffe, A., 1997. "Proof and the Evolution of Mathematics." Synthese 111 (2)=
, 133-146.
Jaffe, A. and F. Quinn, 1993. "Theoretical Mathematics: Towards a Cultural =
Synthesis of Mathematics and Theoretical Physics." Bull. ?AMS 29: 1-13.
----. 1994. Responses to M. Atiyah et al. 1994. Bull. AMS 30: 208-211.
Kitcher, P., 1983. The Nature of Mathematical Knowledge. New York: Oxford U=
niversity Press.
----. 1988. "Mathematical Naturalism", in R. Aspray & P. Kitcher (eds.) His=
tory and Philosophy of Modern Mathematics, Minnesota Studies in the Philoso=
phy of Science, Vol. XI, Minneapolis, University of Minnesota Press (1988),=
 293-323.
Kleiner, I. And N. Movshowitz-Hadar, 1997. "Proof: A Many-Splendored Thing"=
, Mathematical Intelligencer 19 (3): 16-32.
Kolata, G., 1976. "Mathematical Proofs: The Genesis of Reasonable Doubt." S=
cience 192: 989-90.=20
Mazur, B., 1997. "Conjecture." Synthese 111, 197-210.
Moore, E.H., 1902. "A Definition of Abstract Groups." Transactions AMS 3, 4=
85-492
Moore, G.H., 1987. "A House Divided Against Itself: the Emergence of First-=
Order Logic as the Basis for Mathematics." In E.R. Phillips (ed.) Studies i=
n the History of Mathematics, MAA Studies in Mathematics, 98-136.
Peckhaus, V., 1990. Hilbertprogramm und Kritische Philosophie. Des G?ttinge=
r Modell interdisziplin?rer Zusammenarbeit zwischen Mathematik und Philosop=
hie, G?ttingen, Vandenhoeck & Ruprecht..
Rabin, M. 1976. "Probabilistic Algorithms." In Algorithms and Complexity: N=
ew Directions and Recent Results, edited by J.F. Traub, New York: Academic =
Press, 21-40.
Rowe, D., 1998. "Perspectives on Hilbert - A Review Essay", in Perspectives=
 on Science (Forthcoming).
Scanlan, M. 1991. "Who were the American Postulate Theorists?", Jour. Symb.=
 Logic 50, 981-1002.
Spalt, D., 1987. Vom Mythos der mathematischen Vernunft, Darmstadt, Wissens=
chaftliche Buchgesellschaft.
Stegm?ller, W., 1979. The Structuralist View of Theories: A Possible Analog=
ue to the Bourbaki Programme in Physical Sciences, Berlin/New York, Springe=
r.
Thurston, W., 1994. "Letter to the Editor." Scientific American 270 (1), 9.
Toepell, M. 1986. ?ber die Entstehung von David Hilberts =84Grundlagen der =
Geometrie", G?ttingen, Vandenhoeck & Ruprecht.
Tymoczko, T., 1979. "The Four-Color Problem and Its Philosophical Significa=
nce." Journal of Philosophy 76: 67-114.
Tymoczko, T. (ed.), 1985. New Directions in the Philosophy of Mathematics, =
Boston, Birkh?user.
Veblen, O. 1904. "A System of Axioms for Geometry." Transactions AMS 5, 343=
-384.
van der Waerden, B.L., 1930. Moderne Algebra, 2 vols., Berlin, Springer.
Weyl, H. 1944. "David Hilbert and his Mathematical Work." Bulletin AMS 50, =
612-654.
Zeilberger, D., 1994. "Theorems for a Price: Tomorrow's semi-rigorous mathe=
matical culture." Notices of the AMS 40: 978-981 (Reprint: Math Int. 16 (4)=
: 11-14).

1 I wish to thank David Rowe for helpful editorial comments.
2 For a more detailed discussion of this scheme see Corry 1989.
3 Peckhaus 1990.
4 Corry 1998.
5 Corry 1997.
6 The German original is quoted in Toepell 1986, 21. Similar testimonies ca=
n be found in many other manuscripts of Hilbert's lectures. Cf., e.g., Toep=
ell 1986, 58.
7 Quoted from manuscript lecture notes of 1894, in Toepell 1986, 58.
8 See Rowe 1998.
9 The axiom is formulated in Hilbert 1903, 16. Toepell 1986, 254-256, brief=
ly describes the relationship between Hilbert's Vollst?ndigkeit axiom and r=
elated works of other mathematicians. The axiom underwent several changes t=
hroughout the various later editions of the Grundlagen, but it remained cen=
tral to this part of the argument. Cf. Peckhaus 1990, 29-35. The role of th=
is particular axiom within Hilbert's axiomatics and its importance for late=
r developments in mathematical logic is discussed in Moore 1987, 109-122. I=
n 1904 Oswald Veblen introduced the term "categorical" (Veblen 1904, 346) t=
o denote a system to which no irredundant axioms may be added. He believed =
that Hilbert had checked this property in his own system of axioms. See Sca=
nlan 1991, 994.
10 As it is well-known, Kurt G?del proved in 1931 that such a proof is impo=
ssible in the framework of arithmetic itself.
11 Gabriel et al. (eds.) 1980.
12 See Boos 1985.
13 See Corry 1996, 138-147.
14 It is necessary to remark that the term "completeness" used by Hilbert h=
as a different meaning from the one later assigned to this term in the cont=
ext of model theory. Hilbert's idea of completeness of an axiomatic system =
was derived from Hertz's "correctness" of scientific images, and it meant, =
simply, that all the known facts of the theory in question might be derived=
 from the given system of axioms. See Corry 1997.
15 For instance Moore 1902, Huntington 1902.
16 On the American postulationalists and Hilbert's response (or lack of it)=
 to their works, see Corry 1996, =A7 3.5.
17 Quoted in Corry 1996, 162. Unless otherwise noted, all translations into=
 English are mine.
18 Corry 1998.
19 Corry 1998.
20  It is interesting, by the way, that Hilbert's foundational reductionism=
 was expressed in this article in purely logicistic, rather than formalisti=
c, terms. Hilbert mentioned the efforts of Frege and Russell in this direct=
ion and stated that in the eventual completion of Russell's program for axi=
omatizing logic one could recognize the highest achievement of axiomatizati=
on in general ("In der Vollendung dieses gro?z?gigen Russellschen Unternehm=
ens der Axiomatisierung der Logik k?nnte man die Kr?nung des Werkes der Axi=
omatisierung ?berhaupt erblicken"). See Hilbert 1918, 153.=20
21 Hilbert 1916-17, 2-3 (Emphasis in the original): "Fr?her ?bernahm die Ph=
ysik die Lehren der Geometrie ohne weiteres. Dies war berechtigt, solange n=
icht nur die groben, sondern auch die feinsten physikalischen Tatsachen die=
 Lehren der Geometrie best?tigen. Dies war noch der Fall, als Gauss die Win=
kelsumme im Dreieck experimentell mass und fand, dass sie zwei Rechte betr?=
gt. Dies gilt aber nicht mehr von der neuesten Physik. Die heutige Physik m=
uss vielmehr die Geometrie mit in den Bereich ihrer Untersuchungen ziehen. =
Das ist logish und naturgem?ss: jede Wissenschaft w?chst wie ein Baum, nich=
t nur die Zweige greifen weiter aus, sondern auch die Wurzeln dringen teife=
r.
Vor einigen Jahrzehnten konnte man in der Mathematik eine analoge Entwicklu=
ng verfolgen; einen Satz hielt man damals nach Weierstrass dann f?r bewiese=
n, wenn er auf Beziehungen zwischen ganzen Zahlen zur?ckf?hrbar war, deren =
Gesetz man als gegeben hinnahm. Sich mit diesen zu besch?ftigen, wurde abge=
lehnt und den Philosophen ?berlassen. Kronecker sagte einmal: 'Die ganzen Z=
ahlen hat der liebe Gott geschaffen.' Diese waren damals noch einen noli me=
 tangere der Mathematik. Das ging so fort, bis die logischen Fundamente die=
ser Wissenschaft selbst zu wanken begannen. Nun wurden die ganzen Zahlen ei=
nes der fruchtbarsten Arbeitfelder der Mathematik uns speziell der Mengenle=
hre (Dedekind). Der Mathematiker wurde also gezwungen, Philosoph zu werden,=
 weil er sonst aufh=F6rte, Mathematiker zu sein.
So ist es auch jetzt wieder: der Physiker muss Geometer werden, weil sonst =
Gefahr l?uft, aufzuh?ren, Physiker zu sein und umgekehrt. Die Trennung der =
Wissenschaften in F?cher und Fakult?ten ist eben etwas Antropologisches, un=
d der Wirklichkeit Fremdes; denn eine Naturerscheinung fr?gt nicht danach, =
ob sie es mit einem Physiker oder mit einem Mathematiker zu tun hat. Aus di=
esem Grunde d?rfen wir die Axiome der Geometrie nicht ?bernehmen. Darin k?n=
nten ja Erfahrungen zum Ausdruck kommen, die den ferneren Experimenten wide=
rspr?chen."
22 On Bourbaki and the place of the idea of mathematical structures in his =
work, see Corry 1996, Chpt. 7. Readers interested in additional historical =
details and specific references concerning issues discussed in the present =
section will find them in that same chapter.
23 In order to avoid ambiguities I denote by structures (Italics) Bourbaki'=
s technical term (as defined in Volume One of the El?ments. The term withou=
t italics denotes all other, non-formally defined, meanings of the term.
24 This point is discussed in the detail in the above mentioned chapter of =
my book.
25 For more details see Corry 1996, Chpt. 1.
26 See Beaulieu 1993; 1994.
27 As Cartan 1958 [1980], 179, said: "That [a final product] can be obtaine=
d at all [in Bourbaki's meetings] is a kind of miracle that none of us can =
explain."
28 See Part Two of Corry 1996.
29 See, e.g., Grattan-Guiness 1979; Spalt 1987.
30 See, e.g., Corcoran 1980, who describes the book as "important and abiti=
ous," but at the same time regrets that the author does not know enough log=
ic, that his historical claims are inaccurate, and that his philosophocal a=
rguments are unsound.=20
31 Similar views are also put forward in Kitcher 1988, esp. 294-298.
32 For a discussion of the philosophical problems raised by computer-assist=
ed proofs, see Tymoczko 1979 and Detlefsen & Luker 1980. On "very long proo=
fs" see Kitcher 1983, 40 ff. On proofs based on probabilities, see Kolata 1=
976.
33 Kleiner & Movshowitz 1997 contains an attempt to look at this recent deb=
ate from a broader historical perspective (which is, nevertheless, rather d=
ifferent from the one intended in the present article).=20
34 A selection of the lectures presented at this meeting was recently publi=
shed in Synthese, Vol. 111 (2), 1997. On the influence of new media on math=
ematical culture, see Jaffe 1997. On the novel role played by "architectura=
l conjectures" in the construction of new mathematical theories, see Mazur =
1997.=20
[EC1]